Multiplexing of Mixed Numerologies in OFDM/Filtered-OFDM Systems Juquan Mao Submitted for the Degree of Doctor of Philosophy from the University of Surrey Institute for Communication Systems Faculty of Engineering and Physical Sciences University of Surrey Guildford, Surrey GU2 7XH, U.K. November 2019 c Juquan Mao 2019 Summary The flexibility in supporting heterogeneous services with vastly different technical requirements is one of the distinguishing characteristics of the fifth generation (5G) communication systems and beyond. A generic framework is developed in this thesis to address the coexistence/isolation issues of mixing multiple services over a unified physical infrastructure, where the system bandwidth is divided into several bandwidth parts (BWPs), each being allocated a distinct numerology optimized for a particular service. However, multiplexing of mixed numerologies in the same carrier comes at the cost of induced inter-numerology interference (Inter-NI). The Inter-NI can be mitigated by performing additional filtering process on top of orthogonal frequency-division multiplexing (OFDM) waveform for each numerology, namely filtered OFDM or F-OFDM. The additional filtering operation makes transmitted signal better localized in the frequency domain but worse in the time domain, which in turn causes issues within numerology, such as intra-numerology interference (Inter-NI) and filter frequency response selectivity (FFRS). With the developed analysis framework, the problems within numerology and between different numerologies are analyzed, respectively. The issues of Intra-NI and FFRS are firstly analyzed within single numerology. An Intra-NI-free and a nearly-free condition for an F-OFDM system are discussed, and an algorithm on how to select the optimal cyclic redundancy (CR) length is presented. In addition, the impact of FFRS is analyzed for both single antenna and multiple antenna cases, and a pre-equalized F-OFDM (PF-OFDM) system is proposed to tackle the issue. The level of distortion, the Intra-NI and Inter-NI, is quantified by the developed analytical metrics, each of which is a function of several system parameters. Consequently, this leads to an analysis and evaluation of these parameters for meeting a given signal distortion target. A case study utilizing the offered analysis is also presented, where an optimization problem of power allocation is formulated, and a solution is also proposed in multi-numerology systems. It is also demonstrated that a F-OFDM system better addresses the coexistence/isolation problem of mixed numerologies. The work in this thesis provides an insightful analytical guidance for the multi-service design in 5G and beyond systems. Key words: 5G, mixed numerologies, OFDM, intra-numerology interference, internumerology interference, multi-service, power allocation. Email: juquan.mao@surrey.ac.uk WWW: http://www.eps.surrey.ac.uk/ Acknowledgements I would like to express deepest gratitude to my supervisor Prof. Pei Xiao for his full support, expert guidance, understanding and encouragement throughout my study and research. Without his incredible patience and tolerance, my thesis would have been a frustrating pursuit. In addition, I express my appreciation to Dr. Konstantinos Nikitopoulos and Dr. Lei Zhang for having supported my study. Their thoughtful questions and comments were valued greatly. I would also like to gratefully acknowledge the support provided by colleagues and staff at Institute for Communication Systems, home of 5G Innovation Center. Finally, I would like to thank my wife and my family for their unconditional love and support over the years; I would not be able to complete this report without their continuous love and encouragement. Contents List of Notations and Abbreviations i List of Figures ix 1 Introduction 1 1.1 Challenges and State of the Art . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Contribution and Achievements . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Background and Literature Review 2.1 2.2 2.3 OFDM and Its Inspired Waveforms for 5G and Beyond 7 . . . . . . . . . 8 2.1.1 CP-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Waveforms with Subcarrier-Based Filtering . . . . . . . . . . . . 11 2.1.3 OFDM-Based Waveforms with Additional Signal Processing . . 12 OFDM Mixed Numerologies for 5G NR . . . . . . . . . . . . . . . . . . 17 2.2.1 Flexible Numerology and Frame Structure . . . . . . . . . . . . . 17 2.2.2 Multiplexing of Different Numerologies . . . . . . . . . . . . . . . 19 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 A Generic Analysis Model for Multi-numerology OFDM/Filtered OFDM Systems 22 3.1 Transmitter Baseband Processing . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 OFDM Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.2 Transmitter Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.3 Multiplexing of Mixed Numerologies . . . . . . . . . . . . . . . . 28 i Contents ii 3.2 Passing Signal Through the Channel . . . . . . . . . . . . . . . . . . . . 30 3.3 Receiver Baseband Processing . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.1 Filtering at the Receiver . . . . . . . . . . . . . . . . . . . . . . 31 3.3.2 OFDM Demodulation . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.3 Equalization and Detection . . . . . . . . . . . . . . . . . . . . . 33 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 4 Intra-Numerology Interference and Filter Selectivity Analysis 35 4.1 Noise Distribution in F-OFDM Systems . . . . . . . . . . . . . . . . . . 36 4.2 Intra-Numerology Interference Analysis . . . . . . . . . . . . . . . . . . 36 4.2.1 The Expression of the Intra-NI Signal . . . . . . . . . . . . . . . 37 4.2.2 Channel Diagonalization and Intra-NI-free Systems . . . . . . . . 38 4.2.3 The Analytical Expression of the Intra-NI Power . . . . . . . . . 39 4.2.4 Intra-NI Mitigation: A Practical Approach for Choosing CR Length 41 4.2.5 An Alternative for Intra-NI Mitigation: Frequency Domain Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Filter Selectivity Analysis and Discussion . . . . . . . . . . . . . . . . . 44 4.3.1 Filter Selectivity in Single Antenna Systems . . . . . . . . . . . . 45 4.3.2 Filter Selectivity in Multi-Antenna Systems . . . . . . . . . . . . 47 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4.1 Numerical Results for Intra-NI and FFRS . . . . . . . . . . . . . 52 4.4.2 Numerical Analysis for Filter Selectivity . . . . . . . . . . . . . . 59 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3 4.4 4.5 5 Inter-Numerology Interference Analysis 5.1 Inter-Numerology Interference Analysis 63 . . . . . . . . . . . . . . . . . . 64 5.1.1 The Expression of the Inter-NI Signal . . . . . . . . . . . . . . . 64 5.1.2 The Analytical Expression of the Inter-NI Power . . . . . . . . . 66 5.1.3 Further Discussion on Inter-NI . . . . . . . . . . . . . . . . . . . 67 5.2 A Case Study: Power Allocation in the Presence of Mixed Numerologies 68 5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 iii Contents 6 Conclusions and Future Works 80 6.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A Proof of Proposition 3.1 and 3.2 (i) 86 (i) 88 B Derivation of zk C Derivation of z̃k C.0.1 84 (i−) In the case of j ∈ Snum . . . . . . . . . . . . . . . . . . . . . . . (i+) C.0.2 In the case of j ∈ Snum . . . . . . . . . . . . . . . . . . . . . . . . 89 89 D The Proof of Strictly Triangular Matrices 91 Bibliography 93 List of Figures 2.1 High-level 5G use-cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Transmitter structure of OFDM-based waveform for 5G NR . . . . . . . 14 2.3 The comparison of OOB emission. . . . . . . . . . . . . . . . . . . . . . 15 2.4 Nultiplexing numerologies in the frequency domain . . . . . . . . . . . . 19 3.1 System model of OFDM/F-OFDM in the presence of mixed numerologies 24 3.2 Matrix shapes of filter forward/backward spreading . . . . . . . . . . . . 26 3.3 An example of symbol overlap among different numerologies. . . . . . . 28 4.1 Illustration of intra-numerology interference . . . . . . . . . . . . . . . . 37 4.2 Ideal low-pass filters versus practical filters . . . . . . . . . . . . . . . . 46 4.3 Illustration of pre-equalizer. . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 A block diagram of a generic filtered SFBC- OFDM system with two transmit antennas and a single receive antenna. . . . . . . . . . . . . . . 48 4.5 Power of desired signal and intra-NI signal components . . . . . . . . . . 53 4.6 Max, min, and average normalized power of ICI/ISI . . . . . . . . . . . 54 4.7 Average effective interference power . . . . . . . . . . . . . . . . . . . . 55 4.8 Error performance for F-OFDM systems under AWGN channel . . . . . 57 4.9 Error performance comparison with and without implementation of BwPIC for FOFDM systems under AWGN channels with QPSK modulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.10 Interference power (normalized by signal power) versus filter frequency responses of consecutive subcarriers. . . . . . . . . . . . . . . . . . . . . 59 4.11 Error performance comparison with and without implementation of preequalization under AWGN channels with QPSK modulation. . . . . . . 60 4.12 BER performance for filtered SFBC-OFDM systems with and without pre-equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 iv List of Figures v 5.1 Inter-numerology interference. . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2 SIR with different guard band settings on interfering BWPs . . . . . . . 73 5.3 SIR with different settings on the length of filters of interfering sources . 74 5.4 SIR with different settings on the length of filters of interfering sources . 75 5.5 BER performance with different settings on power offset . . . . . . . . . 77 5.6 Spectrum efficiency comparison among different power allocation schemes 78 List of Notations and Abbreviations List of Abbreviations 4G fourth-generation 5G fifth-generation ACI adjacent carrier interference AWGN addictive white Gaussian noise b-ISI backword inter-symbol interference BER bit error rate BPSK binary phase shift keying BwPIC block-wise parallel interference cancellation BWP bandwidth part CP cyclic prefix CR cyclic redundancy CSI channel state information CS cyclic suffix DFT-S-OFDM discrete Fourier transform spread OFDM eMBB enhanced mobile broadband ETU extended typical urban vi LIST OF NOTATIONS AND ABBREVIATIONS f-ISI forward inter-symbol interference FBMC filter-band multi-carrier FFT fast Fourier transform GB guard band GFDM frequency-division multiplexing GI guard interval IC interference cancellation IFFT inverse fast Fourier transform Inter-NI inter-numerology interference Intra-NI intra-numerology interference ISI inter-symbol interference ITU International Telecommunication Union LTE long term evolution M2M machine-to-machine MIMO multiple-input multiple-output MMSE minimum mean square error mMTC massive machine type communications NR new radio OFDM frequency-division multiplexing OOB out-of-band PAPR peak-to-average power ratio PA power amplifier PWD power spectrum distribution QAM quadrature amplitude modulation QoS quality of service vii LIST OF NOTATIONS AND ABBREVIATIONS Rx receiver SC-CPS single carrier circularly pulse shaped SC-FDE single-carrier frequency domain equalization SE spectrum efficiency SFBC space frequency block coding SFMC subband filtered multi-carrier SIC successive interference cancellation SINR signal-to-interference and noise ratio SIR single to interfernece ratio STBC space-time block code TDD time domain duplexing Tx transmitter UE user equipment UFMC universal filtered multi-carrier uRLLC ultra-reliable and low-latency communications V2V vehicle-to-vehicle W-OFDM windowed OFDM WOLA weighted overlap-and-add ZF zero-forcing viii Numerology-ralated Notations (i) a single-numerology signal in the k-th OFDM window of the i-th numerology (i←j) a signal from j-th numerology captured in the k-th OFDM window of the i-th numerology xk xk LIST OF NOTATIONS AND ABBREVIATIONS ix x<i> k a mixed signal from all numerologies which captured in the k-th OFDM window of the i-th numerology x(i) , x(i) , X(i) indicate variables only related to the i-th numerology Notations (·)∗ complex conjugate operator E{·} expectation function (·)H conjugate transpose operator (·)T transpose operator C complex space R real space X boldface upper-case characters represent matrices x boldface lower-case represent vectors IN N dimensional identity matrix blkdiag(A, n) returns a column vector of the main diagonal elements of matrix A diag(A) returns a column vector of the main diagonal elements of matrix A diag(x) returns a square diagonal matrix with the elements of vector x on the main diagonal Chapter 1 Introduction Mobile networks have been evolving to better interconnect people, and approximately every 10 years a new generation of mobile technologies is introduced which delivers a significant improvement in performance, efficiency and capability. While the first four generations of mobile networks, from the first-generation to the fourth-generation (4G), focusing on interconnecting people by offering better voice and mobile broadband data services, the fifth-generation (5G) promises to deliver a fully mobile and connected society, i.e., not only connecting people but also providing machine type communications and a broad range of services with disparate requirements. The services supported by 5G have diversified variance of requirements on coverage, throughput, capacity, latency, and reliability. Accordingly, the International Telecommunication Union (ITU) has categorized 5G services into three main usage scenarios [1]: enhanced mobile broadband (eMBB), ultra-reliable and low-latency communications (uRLLC), and massive machine type communications (mMTC). Each of these scenarios demands distinct quality of service (QoS) requirements, such as throughput, latency, reliability, and number of connected users/devices. 1.1 Challenges and State of the Art The coexistence of aforementioned services with such diverse requirements poses challenges to legacy one-size-fits-all radio systems, such as the traditional 4G long term 1 1.1. Challenges and State of the Art 2 evolution (LTE) mobile networks, which is designed to meet requirements of conventional MBB services with a single orthogonal frequency-division multiplexing (OFDM) numerology1 . The one-fit-all structure may not be sufficiently flexible to meet all envisioned 5G use cases [3, 4]. For instance, an mMTC service requires smaller frequency subcarrier spacing (thus longer symbol duration) to support massive delay-tolerant devices and to provide power boosting gain in some extreme cases, while vehicleto-vehicle (V2V) communications necessitate significantly larger frequency subcarrier spacing (thus smaller symbol duration) for stringent latency requirements and more robustness to Doppler spread. Considering that the multitude of heterogeneous services must be provided simultaneously over a unified underlying physical layer, 5G new radio (NR) adopts a set of numerologies to suit different technical requirements and frequency bands [5]. The multiplexing of different numerologies can be implemented either in the time or the frequency domain. The latter has better compatibility and support for multi-service coexistence in comparison to the time domain counterpart [6]. However, multiplexing of different numerologies in the frequency domain inevitably introduces interference between numerologies [7] due to the fact that the subcarrier orthogonality possessed by single numerology no longer holds [8]. The interference, which refers to internumerology interference (Inter-NI), is high in conventional OFDM waveform [6] due to its poor out-of-band (OOB) emission property. A natural solution to reduce the Inter-NI is to insert sufficient guard band (GB) between numerologies. However, it comes at a cost of degraded spectrum efficiency. Alternatively, new waveforms with better spectral localization property can have a significant impact on the Inter-NI. Motivated by the above mentioned considerations, various waveforms [9–16] have been proposed to reduce OOB emission. An overview and a comprehensive comparison among these waveforms in term of qualitative and quantitative analysis can be found in [2] and [17]. Considering the performance-complexity trade-off, multiple-input multipleoutput (MIMO) friendliness, forward/backward compatibility, 3GPP [18] has decided that the base waveform is still cyclic prefix OFDM (CP-OFDM) in 5G NR, and some 1 Numerology refers to configuration of waveform parameters, such as subcarrier spacing/symbol duration and cyclic prefix in OFDM [2]. 1.2. Objectives 3 spectral confinement techniques, such as filtering or windowing, can be employed on top of the base waveform to reduce OOB emission. The flexibility in choosing these techniques is given to device manufactures, provided that the added filtering/windowing is transparent between transmitter (Tx) and receiver (Rx). When filtering/windowing is applied, the OFDM symbol duration expands in the time domain as frequency localization improves. In the case when the duration extends over the coverage of cyclic redundancy, the orthogonality among subcarriers in the same numerology is destroyed and intra-numerology interference (Intra-NI) occurs. Interference analysis for multi-numerology systems has been attracting an increased interest recently. In particular, [19–21] discussed the factors contributing to interference in windowed OFDM (W-OFDM) systems. Moreover, a framework for subband filtered multi-carrier (SFMC) systems is introduced in [8], and the interference of universal filtered multi-carrier (UFMC) systems is also analyzed in the presence of transceiver imperfections and insufficient guard interval between symbols. In [22], the authors report a field trial conducted on a configurable testbed in a real-world environment for the performance evaluations of OFDM-based 5G waveform candidates, such as CPOFDM, W-OFDM, and F-OFDM. Their field trial results confirm the feasibility of a single physical layer multi-numerology systems. 1.2 Objectives While both filtering and windowing enable frequency domain multiplexing of mixed numerologies, we focus on the filtering approach in this thesis since it gives a better performance in terms of frequency localization and interference mitigation [17]. To the best of our knowledge, analytical study accounting for all factors contributing to interference in multi-numerology F-OFDM systems is still lacking. Such a study plays a pivotal role in providing guidance on system design for the coexistence and isolation of multiple services, enabling the development of efficient interference cancellation techniques, facilitating the formulation of optimization problems for maximizing spectrum efficiency, and so on. These motivate us to fill the gap and set our objectives of this research as 1.3. Contribution and Achievements 4 • To develop an analytical framework for multi-numerology systems to address the issues on the coexistence and isolation of multiple services over a unified physical layer . • To model and formulate the process of multiplexing of mixed numerologies in the frequency domain. • To analyze the Inter-NI and investigating interference mitigation schemes. • To analyze the Intra-NI and filter frequency response selectivity (FFRS) induced by additional filtering process on top of the OFDM waveform. • To proposing a novel power allocation scheme to maximize system sum-rate for multi-numerology systems. 1.3 Contribution and Achievements The contributions of the thesis are summarized as following: • A generic analytical multi-numerology system model is first developed for OFDM/ F-OFDM systems to address the issue of mixed numerologies coexistence, in which the process of multiplexing different numerologies is formulated. Within the proposed model, all linear convolution operations, such as filtering at the transmitter, passing the channel, and filtering at the receiver, are represented in matrix forms to facilitate the analysis of inter/intra-numerology interference. • The proposed model allows us to divide the intra-numerology interference in FOFDM systems into inter-carrier interference (ICI), forward inter-symbol interference (f-ISI), and backward inter-symbol interference (b-ISI), so that the impact of each interference component can be studied individually. The conditions to achieve Intra-NI-free and nearly-free F-OFDM systems are derived accordingly. An optimization problem on how to choose the size of cyclic redundancy for balancing system efficiency and receiver complexity is then formulated. Furthermore, we propose a novel low-complexity block based parallel interference cancellation 1.3. Contribution and Achievements 5 algorithm based on the well channelized signal from the proposed model for suppressing the Intra-NI. • The system performance degradation imposed by FFRS is firstly analyzed in single antenna F-OFDM systems. The results are extended to the multi-antenna F-OFDM case, and a system model for analyzing spatial orthogonality in multiantenna space frequency block coding (SFBC) F-OFDM systems is developed. In the presence of FFRS, the spatial orthogonality is proved to be invalid and the analytical expression of spatial interference power is derived. A pre-equalizer is proposed at the transmitter to alleviate the adverse effect of the interference at a cost of subband bandwidth-dependent power loss. • Given the developed analytical framework, the interference between different numerologies is analyzed, and a metric to quantify the distortion level is derived as a function of several system parameters. This enables the analysis of the impact from each of these parameters, which leads to a more accurate and insightful approach for system design than simulation-based models. • Based on the aforementioned analysis, a case study on optimizing power allocation is presented, where a problem is formulated and solved in multi-numerology systems. The research carried out in this thesis results in the following outcomes: 1. J. Mao, L. Zhang, P. Xiao, and K. Nikitopoulos, “Intrinsic In-band Interference and Filter Frequency Response Selectivity Analysis for Filtered OFDM Systems,” IEEE Transactions on Signal Processing, under review. 2. J. Mao, L. Zhang, P. Xiao, and K. Nikitopoulos, “Interference Analysis and Power Allocation in the Presence of Mixed Numerologies,” IEEE Transactions on Wireless Communications, second-round review. Other waveform-related publications: 1.4. Overview of Thesis 6 1. J Mao, C Wang, L Zhang, et al. ”A DHT-based multicarrier modulation system with pairwise ML detection.” In 2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), pp. 1-6. IEEE, 2017. 2. J. Zheng, J Mao, et al. ”Iterative frequency domain equalization for MIMOGFDM systems” IEEE Access 6 (2018): 19386-19395. 3. C. He, L. Zhang, J. Mao,et al. ”Performance analysis and optimization of DCTbased multicarrier system on frequency-selective fading channels.” IEEE Access 6 (2018): 13075-13089. 4. L. Zhang, A. Ijaz, J. Mao, et al. ”Multi-service signal multiplexing and isolation for physical-layer network slicing (pns).” In 2017 IEEE 86th Vehicular Technology Conference (VTC-Fall), pp. 1-6. 2017. 5. L. Zhang, C. He, J. Mao, et al. ”Channel estimation and optimal pilot signals for universal filtered multi-carrier (UFMC) systems.” In 2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), pp. 1-6. 2017. 1.4 Overview of Thesis The reminder of the thesis proceeds as follows: Chapter 2 discusses the background and the existing works in the context of waveforms and mixed numerologies for addressing multi-service challenges. Chapter 3 presents a generic F-OFDM/OFDM transceiver structure in the presence of mixed numerologies and describes the system model. The issues inflicted by additional filtering operation on top of CP-OFDM within numerology are investigated in Chapter 4, while the interference between difference numerologies is studied in Chapter 5, followed by a case study on optimizing power allocation in multi-numerology systems. The conclusions of the work conducted so far and the plan of the future work are given in Chapter 6. Chapter 2 Background and Literature Review The evolution of mobile communications systems toward the so-called fifth generation (5G) faces the challenging of meeting requirements of Mobile BroadBand (MBB) use cases as well as new ones associated with customers of new market segments and vertical industries (e.g., e-health, automotive, energy). Therefore, in addition to supporting the evolution of the current business models, 5G expands to new ones. The International Telecommunication Union (ITU) has categorized 5G services into three main usage scenarios [1] (see also Fig. 2.1): • Enhanced mobile broadband (eMBB), requiring very high data rates and large bandwidths • Ultra-reliable low-latency communications (URLLC), requiring very low latency, and very high reliability and availability • Massive machine type communications (mMTC), requiring low bandwidth, high connection density, enhanced coverage, and low energy consumption at the user end. The coexistence and isolation of heterogeneous services over a unified underlying physical infrastructure are the two challenges to the 5G NR. To allow the coexistence, a set 7 8 2.1. OFDM and Its Inspired Waveforms for 5G and Beyond eMBB High data rates, high traffic volume 5G mMTC uRLLC Massive Number of devices, low cost, low energy consumption Very Low Latency, vey high reliability and availability Figure 2.1: High-level 5G use-cases of numerologies optimized for different technical requirements [5], are multiplexed in frequency or time domain within one baseband. To better isolate services (avoid nontrivial interference among services), new waveforms with low out-of-band emission are being investigated. Therefore, numerologies and waveforms are critical to multi-service networks. An overview of potential 5G waveforms and numerologies is presented in the next sections. 2.1 OFDM and Its Inspired Waveforms for 5G and Beyond On the road to 5G NR, numerous waveforms have been proposed which can be roughly categorized into three groups: the classic CP-OFDM, waveforms with subcarrier-based filtering, and OFDM-based waveforms with additional signal processing. An overview will be given in the sequel. 2.1. OFDM and Its Inspired Waveforms for 5G and Beyond 2.1.1 9 CP-OFDM OFDM, as the most popular solution to combat inter-symbol interference (ISI), has enjoyed its dominance in many wired [23] [24] and wireless systems [25] [26]. It has been adopted in different variations of digital subscriber line (DSL) standards, as well as in most of wireless standards, e.g., variations of IEEE 802.11 and IEEE 802.16, the third generation partnership program long-term evolution (3GPP-LTE), and LTEAdvanced. High-data-rate transmission is demanded by many applications in communication systems. However, the symbol duration reduces as the data rate increases, and dispersive fading of the wireless channels will cause more severe ISI if single-carrier modulation is used. To reduce the effect of ISI, the symbol duration must be much larger than the delay spread of wireless channels [27]. In OFDM [28], the entire channel is divided into many narrow-band sub-channels, which are transmitted in parallel to maintain high-data-rate transmission and, at the same time, to increase the symbol duration to combat ISI. It has many advantages [29] and is a perfect solution for point-to-point and downlink transmissions. • Spectrum efficiency: A significant advantage of OFDM is improved spectrum efficiency by using closely-spaced overlapping subcarriers. • Robustness to channel selective fading: OFDM is more resilient to frequency selective fading than single carrier systems as it divides a wide band into multiple narrow bands in each of which transmitted signal experiences flat fading. • Time localization: OFDM is well-localized in time domain, which is important to efficiently enable time domain duplexing (TDD) and support latency critical applications. • Resilience to ISI: Another advantage of OFDM is its strong resilience to ISI due to the low data rate on each of the subcarrier. • Low transceiver complexity: OFDM enjoys its easy implementation where each OFDM symbol synthesized as a summation of number of subcarriers (complex- 2.1. OFDM and Its Inspired Waveforms for 5G and Beyond 10 valued sinusoidal signals) that are modulated by a single operation - inverse fast Fourier transformation (IFFT). • Simple channel equalization: Channel equalization becomes much easier as can be performed independently on each flat-fading sub-channel. However, OFDM is challenged in many ways when applied to more complex networks. Recently, it has became a consensus that the basic waveform of 5G should at least be able to offer: 1. Tailored services to different needs and channel characteristics, 2. Reduced out-of-band emission (OOBE), 3. Extra tolerance to time-frequency misalignment [30]. OFDM appears insufficient to meet above requirements. For instance, to guarantee orthogonality and thus avoid inter-symbol/carrier interference, stringent time and frequency alignment is required, resulting in heavy signaling for synchronization, especially for uplink transmissions. Such synchronization has been found very difficult to establish, especially in highly mobile environments where Doppler shift/spread are not easy to track. The authors in [31] have stated that “carrier and timing synchronization represents the most challenging task in OFDMA systems.” To address the issue, several methods have been proposed [32–34]. These methods are generally very complex to implement and the receiver complexity increases by orders of magnitude [35]. These solutions may be too slow and/or too resource intensive to be deployed in many applications especially for low cost devices in 5G. Another weakness of OFDM appears when transmission is carried out over a set of non-contiguous frequency bands. The poor response of the sinc filters in IFFT/FFT introduces OOB egress noise to other users and also collects significant ingress noise from them. The same problem appears if one attempts to adopt OFDM to fit in the spectrum holes in cognitive radios. Methods of reducing OFDM spectral leakage, aka., OOB emissions, have been proposed in [36–38] to tackle the issue, however the the 2.1. OFDM and Its Inspired Waveforms for 5G and Beyond 11 performance is very limited with an OOB emission suppression of only 5 to 10 dB at a cost of reduced bandwidth efficiency and significantly increased complexity to the Tx. Motivated by the above mentioned issues, various waveforms such as filter-band multicarrier (FBMC), generalized frequency-division multiplexing (GFDM), W-OFDM, universal filtered multi-carrier (UFMC), and filtered OFDM (F-OFDM) have been proposed. Among them, FBMC and GFDM are subcarrier-based filtering, where a prototype filter is shifted and applied to each single carrier, while the others are OFDM-based waveform. These waveforms will be presented in the sequel. 2.1.2 Waveforms with Subcarrier-Based Filtering FBMC [9, 10] applies filtering on a per-subcarrier basis and is considered as an attractive alternative to OFDM to provide improved OOB spectrum characteristics. Since subcarrier filters are narrow in frequency and thus require long filter lengths (normally at least to preserve an acceptable ISI and ICI), the symbols are overlapped in the time domain. To comply with the real orthogonality principle, offset-QAM (OQAM) can be applied and, therefore, FBMC is not orthogonal in the complex domain. Owing to the very low OOB emission of the subcarrier filters, the synchronization requirement of FBMC in the uplink of multi-user networks is largely relaxed [39]. In the cognitive radios, the filter bank for OOB emission can also be used for spectrum sensing [30,40,41]. On the other hand, in comparison to OFDM, FBMC falls short of compatibility with MIMO techniques, although a few attempts to combine FBMC with MIMO channels have been studied in [42], [43], and the research in this domain is still immature. Because the subcarriers have narrow bandwidth, the length of the transmit filter impulse response is usually long. Typically, the filter has four times the length of the symbols. Clearly, this solution is not suitable for low latency scenarios, where high efficiency must be achieved for short burst transmissions. Unfortunately, this can be problematic for machine-to-machine (M2M) communications and Internet of Things (IoT) in 5G as it involves transmission of very short messages. A flexible multicarrier modulation scheme, namely GFDM, was first presented by Fettweis et al. in 2009 [44]. The flexibility of GFDM allows it to cover CP-OFDM and 2.1. OFDM and Its Inspired Waveforms for 5G and Beyond 12 single-carrier frequency domain equalization (SC-FDE) as special cases. It is based on the modulation of independent blocks, where each block consists of a number of subcarriers and subsymbols. The subcarriers are filtered with a prototype filter that is circularly shifted in time and frequency domain to reduce the OOB emissions. Consequently, the tight synchronization requirement in multi-user scenarios is relaxed. Based on the developed framework, Fettweis and his team investigated further on other aspects of GFDM [45–50]. We also investigated the issues when GFDM is implemented with MIMO transmission and proposed an novel equalization method in [51]. Specifically, GFDM is a block-based multicarrier transmission scheme capable of spreading data across a two-dimensional (time and frequency) block structure (multi-symbols per multi-carriers). The block based transmission scheme enables one CP for several GFDM symbols so that the overhead can be reduced and thus bandwidth efficiency is improved. The drawback of GFDM is that the orthogonality among subcarriers no longer holds due to the subcarrier filtering, and both ISI and ICI might arise. This inevitably causes additional decoding complexity to the receiver and may incur performance degradation. The term non-orthogonal waveform is often used to describe this property of GFDM [11]. More recently, the concept of circularly convolved filtering based on which GFDM waveform is built has been used by FBMC [52–55]. The waveform that results from this change refers to circular FBMC (C-FBMC) or OFDM/C-OQAM. C-FBMC reduces the signal overheads, while preserving the real-orthogonality of the prototype filter, and as such, does not incur any ISI/ICI on the demodulated OQAM symbols. Another interesting benefit of C-FBMC waveform is its better compatibility with MIMO. Some work which extend GFDM/C-FBMC to MIMO channels can be found in [11] [56]. 2.1.3 OFDM-Based Waveforms with Additional Signal Processing Universal Filtered Multicarrier (UFMC) [14, 57–59] is another candidate waveform where a group of subcarrier is filtered to reduce the OOB emission. Because the bandwidth of the filter covers several subcarriers, its impulse response can be short, which means that high spectral efficiency can be achieved for short burst transmissions. 13 2.1. OFDM and Its Inspired Waveforms for 5G and Beyond Table 2.1: Comparison of 5G potential waveforms [2, 60]. Performance indicator CP-OFDM FBMC GFDM UFMC Spectral efficiency High High Hign Hign Time localization High Low Low Low PAPR High Medium Medium Hign MIMO compatibility High Low Medium Low Medium Low Low Medium Complexity Low High High Hign Flexibility High High High Hign Out-of-band emissions High Low Medium Low Robust. to freq. selective chan. High High High Hign Robust. to time selective chan. Medium Medium Medium Medium Phase noise robustness UFMC does not require a CP and it is possible to design the filters to obtain a total block length equivalent to the CP-OFDM. However, because there is no CP, UFMC is more sensitive to small time misalignment than CP-OFDM [14]. Hence, UFMC might not be suitable for applications that require loose time synchronization to save energy. Up to now, we have seen that the conventional OFDM has been tweaked in any possible way, e.g., subcarrier-wise filtering or pulse shaping, filtering of groups of sub-carriers, allowing successive symbols to overlap in time, dropping cyclic-prefix, replacing cyclicprefix with nulls or with another sequence. These attempts address some challenges that future communications face, however none of them can satisfy all the requirements and each of them brings new problems such as receiver complexity, non-compatibility with MIMO, and increased latency, etc. A summary of qualitative comparisons of these waveforms is given in Table 2.1. To sum up, CP-OFDM still ranks the best in terms of the performance indicators that matter most: compatibility with multi-antenna technologies, high spectral efficiency, and low implementation complexity. Moreover, CP-OFDM is well-localized in time domain, which is important for latency critical-applications and TDD deployments. The drawbacks of poor frequency localization and high peak-to-average power ratio (PAPR), can be resolved by additional signal processing before and/or after CP-OFDM modulation. According to the recent agreement reached by 3GPP [18], OFDM is still 14 2.1. OFDM and Its Inspired Waveforms for 5G and Beyond Coded bits Symbol Mapping S/P Precoding IFFT CP P/S Filtering To RF CP-OFDM essentials Post/pre signal processing Windowing Figure 2.2: Transmitter structure of OFDM-based waveform for 5G NR the base of the new waveform for 5G, additional filtering, windowing or precoding, are considered to achieve the better PAPR or frequency localization as illustrated in Fig.2.2 [61]. The comparison of OOB emission among all mentioned waveforms is found in Fig. 2.3 [17]. Precoding: A linear processing of input data before IFFT is usually known as precoding, and may be helpful to improve OOB emission and PAPR. One representative example is discrete Fourier transform spread OFDM (DFT-S-OFDM) waveform that has been adopted in LTE uplink transmissions due to its low PAPR. Numerous variants of DFT-S-OFDM have been proposed for NR [62–65]. Zero-tail DFT-S-OFDM [62] aims at omitting CP by letting the tail samples approximate to zero. Guard interval (GI) DFT-S-OFDM [63] superposes a Zadoff-Chu sequence to the tail samples for synchronization purposes. Unique word DFT-S-OFDM replaces zeros in front of the DFT by certain fixed values to adaptively control waveform properties. On the other hand, single carrier circularly pulse shaped (SC-CPS) and generalized precoded OFDMA waveforms [64] use pre-specified frequency domain shaping after the DFT for further PAPR reduction at the cost of excess bandwidth. CPS-OFDM may be regarded as a generalized framework that flexibly supports multiple shaped subcarriers in a subband. DFT-S-OFDM-based waveforms, in contrast to filter-based waveforms, usually make 2.1. OFDM and Its Inspired Waveforms for 5G and Beyond 15 Figure 2.3: The comparison of OOB emission. W-GFDM is windowed GFDM in which additional windowing is applied to GFDM to soften the transition between adjacent blocks and thus reduce the OOBE. UF-OFDM is another name of UFMC. it much easier to maintain linear operation for power amplifier (PA) with less deterioration from lowering OOBE. Moreover, an appropriate modification of modulation schemes, such as π/2 binary phase shift keying (BPSK) [65], can greatly assist such waveforms in achieving an extremely low PAPR. Windowing: Windowing is to prevent steep changes between two OFDM symbols so as to confine OOBE. Multiplying the time domain samples residing in the extended symbol edges by raised-cosine coefficients is a widely used method in W-OFDM and weighted overlap-and-add (WOLA) OFDM waveforms [66]. The authors in [13] investigated the performance of W-OFDM in the presence of carrier frequency offset and timing offset, the study revealed that W-OFDM offers robustness to asynchronism. On the other hand, great efforts have been dedicated to improve the performance and flexibility for W-OFDM. Window functions were discussed and optimized in [67]. The receiver windowing was considered along with the Tx windowing to reject the adjacent channel interference and limit the OOB emission, respectively. To alleviate the 2.1. OFDM and Its Inspired Waveforms for 5G and Beyond 16 ISI induced by the reduced CP length while maintain the OOB emission suppression, a time-asymmetric windowing scheme was proposed in [66]. The authors of [68] proposed the windowing scheme for optimal time-frequency concentration for W-OFDM systems. A flexible windowing method was proposed in [69] and further extended in [70] to balance the OOB emission and robustness against channel delay spread. Filtering: Filtering is a straightforward way to suppress OOB emission with a better spectrally-localized filter while enjoying all the benefits of CP-OFDM. The waveform is refered to as F-OFDM. This was attained by allowing the filter-length to exceed the CP length and designing the filter properly. Many aspects of F-OFDM such as general framework and methodology, design and implementation, field trials have been reported in the literatures [22, 71–78]. Most of these works only focus on the the advantages of F-OFDM which are obtained at the cost of sacrificing other performance metrics. For suppressing OOB emission, the filters employed in F-OFDM systems are normally very long (up to half of FFT size [72, 74]), which inevitably makes the systems more prone to the in-band interference. Most of the existing work indicates that it has a trivial influence to system performance for medium to wide subband based on qualitative analysis and simulation with specific parameters. Zhang et al. derived a system model in [16] to quantitatively analyze interference in F-OFDM systems, in which the channel matrix is divided into three parts in order to decompose the total inferences into ISI and adjacent carrier interference (ACI). The limitation of this method is that it can neither differentiate the interference induced by channel or filtering, nor the intercarrier interference from other subbands or its own band. To break the limitation, a new matrix-form system model to enable quantitative analysis of intra-numerology interference is proposed in this thesis. FFRS which refers to the uneven weights of filter frequency response in transition band is also discussed in the thesis. As mentioned earlier, 5G NR recommendation of the waveform for below 52.6 GHz communications is still CP-OFDM and additional signal processing can be conducted on top of CP-OFDM provided that the Tx processing is transparent to the Rx. This implies that any additional signal processing on top of the commonly agreed baseline CP-OFDM waveform, for example, time domain windowing or bandwidth part filtering performed in the Tx, is not signaled to the Rx and thus generally unknown. The 2.2. OFDM Mixed Numerologies for 5G NR 17 feasibility of transparent waveforming is studied in [79]. The flexibility to choose windowing or filtering or none of them is given to manufactures. The comparison between W-OFDM , F-OFDM and CP-OFDM can be found in [22]. In this thesis, the focus is on the study of the filtering instead of windowing. 2.2 OFDM Mixed Numerologies for 5G NR The coexistence of services with diverse requirements poses challenges to legacy onesize-fits-all radio systems, such as the traditional 4G LTE mobile networks, which have been designed to meet requirements of conventional MBB services with a single OFDM numerology. The one-fit-all structure may not meet the diverse requirements of all envisioned 5G use cases. Since the multitude of heterogeneous services should be provided simultaneously by a common underlying physical layer, and separate radio design for each service is not practical due to the unfeasible cost and complexity. In addition, it is cumbersome to design a one-fits-all solution to meet all service requirements [8]. 2.2.1 Flexible Numerology and Frame Structure To support heterogeneous services with different technical requirements over a unified underlying physical layer, 5G new radio (NR) adopts a set of scalable numerologies [5]. A numerology is defined by subcarrier and CP overhead. This additional degree of freedom offers more flexibility in supporting multi-service. For instance, a larger subcarrier spacing (shorter OFDM symbol) is more suitable for high Doppler and low latency cases, while a smaller subcarrier spacing (longer OFDM symbol) is more beneficial for extended coverage and high channel spreading scenarios. OFDM numerologies are derived from a baseline OFDM numerology with 15 KHz subcarrier spacing via incorporating a scaling factor. The numerology for higher subcarrier spacings then can be derived by scaling the baseline numerology by power of two. In essence, an OFDM symbol is split into two OFDM symbols of the next higher numerology. The OFDM numerologies supported in 5G NR are given by Table 2.2.1 where µ is a scaling factor. Scaling by power of two is beneficial as it maintains the symbol 18 2.2. OFDM Mixed Numerologies for 5G NR Table 2.2: Supported transmission numerologies by 5G NR µ ∆f = 2µ · 15[kHz] Cyclic prefix # slots per subframe 0 15 Normal 1 1 30 Normal 2 2 60 Normal, Expected 4 3 120 Normal 8 4 240 Normal 16 boundaries across numerologies, which simplifies mixing different numerologies on the same carrier. The features of the 5G flexible numerology can be summarized below. • Subcarrier spacing is no longer fixed to 15 kHz. Rather, the subcarrier spacing scales by 2µ × 15 kHz to suit different technical requirements of the 5G use cases. • Number of slots increases as numerology (µ) increases. Same as LTE, with normal CP, each slot has 14 symbols. Since the duration of OFDM symbol is inverse of subcarrier spacing, the number of slots in one subframe increases with µ; Therefore there are more number of symbols for a given time. • Multiplexing of different numerologies. Different numerologies can be transmitted on the same carrier frequency with a new feature called bandwidth parts (BWP). They can be multiplexed in the frequency domain. Mixing different numerologies on a carrier can cause interference to subcarriers of another numerology. While this provides the flexibility for diverse services to be accommodated on the same carrier frequency, it also introduces new challenges with interference between the different services. Flexible numerology in 5G is much different from numerology found in 4G. It brings more flexibility and degrees of freedom to maximize the utilization of radio resources, but it also introduces new challenges on the way waveforms are designed and employed, 19 2.2. OFDM Mixed Numerologies for 5G NR Overall System Bandwidth BWP1 Numerology 1 BWP2 Numerology 2 BWP3 Numerology 3 Figure 2.4: Nultiplexing numerologies in the frequency domain especially on how to deliver an efficient multiplexing of different numerologies to achieve a better isolation of those services it carries. We will give a overview of the state-ofthe-art on this topic in the next subsection. 2.2.2 Multiplexing of Different Numerologies The multiplexing of different numerologies can be implemented either in the time or the frequency domain. For the conventional CP-OFDM waveform, which is well-localized in time domain, arranging numerologies in time domain can maintain the orthogonality between the consecutive blocks [80]. However, multiplexing services in frequency domain have better forward compatibility and inclusive support for services with different latency requirements compared with the time domain counterpart [8]. Thus, a viable and mostly accepted way to cater for diverse services is to divide the bandwidth into several bandwidth parts (BWPs), as illustrated in Fig. 2.4, and each of them is assigned a different numerology [22]. Multiplexing different numerologies in the frequency domain has been commonly accepted in 3GPP [5]. In the presence of mixed numerologies, subcarrier orthogonality maintains only within a numerology in an OFDM-based system. Subcarriers from one numerology interfere with those from others since subcarriers of one numerology may pick up leaked energy from subcarriers of other numerologies. The OFDM sinc-like transfer function decays with frequency f as slow as 1/f and substantial interference occurs between numerologies [6]. The interference becomes more severe in the presence of power offset between subcarriers in aggressor numerologies and those in a victim numerology. Therefore, multiplexing different numerologies in the frequency domain inevitably comes at a cost 2.2. OFDM Mixed Numerologies for 5G NR 20 of system performance degradation in terms of spectrum efficiency, scheduling flexibility, and computational complexity [7]. In addition, OOB emission from one numerology to the other destroys the subcarrier orthogonality existing in single numerology and inflicts interference between different numerologies, due to the fact that the subcarrier orthogonality possessed by single numerology no longer holds [8]. In order to mitigate Inter-NI, guard bands (GB) can be inserted between BWPs. However, it comes at a cost of spectrum efficiency reduction. Alternatively, new waveforms aiming for lower OOB emission discussed in the previous section of this chapter fit the purpose. Obviously the selected waveform has a critical impact on the Inter-NI. Interference analysis for multi-numerology systems has been attracting an increased interest recently. In particular, [19–21] discussed the factors contributing to interference in W-OFDM systems. In [19], a system model for the interference analysis was established, in which the analytical expression of InterNI power was derived as a function of several parameters in connection to the channel, guard band, and windowing. In [20] and [21], the authors investigated the impact on the power level of interference from guard interval in the time domain and guard band in the frequency domain, respectively, in order to improve bandwidth efficiency. A framework for subband filtered multi-carrier (SFMC) systems was introduced in [8], and the interference of UFMC systems was also analyzed in the presence of transceiver imperfections and insufficient guard interval between symbols. In [79], the feasibility of transparent Tx and Rx waveform processing in mixed-numerology systems was discussed. In [22], the author reported a field trial conducted on a configurable test bed in a real-world environment for the performance evaluations of OFDM-based 5G waveform candidates, such as CPOFDM, W-OFDM, and F-OFDM, and the field trial results confirm the feasibility of multi-numerology systems. While both filtering and windowing enable frequency domain multiplexing of mixed numerologies, the filtering approach is the focus of this thesis since it gives a better performance in terms of frequency localization and interference mitigation [17]. 2.3. Summary 2.3 21 Summary A multi-service system is an enabler to flexibly support diverse communication requirements for the next generation wireless communications. In such a system, multiple services coexist in one baseband system with each requiring its optimal frame structure and low OOB emission waveforms operating on the frequency band to reduce the mutual interference. In other words, the coexistence and isolation of multiple services over a unified physical layer are the issues to address. In this chapter, we have presented the efforts to address the issues from the perspective of waveform and numerology . To sum up, the viable solution is to divide the system bandwidth into several BWPs, each having a distinct numerology optimized for a particular service, and additional signal processing on top of CP-OFDM is employed to better isolate different services. Chapter 3 A Generic Analysis Model for Multi-numerology OFDM/Filtered OFDM Systems The baseline physical layer defination and numerology follow the ones defined in [18]. To accommodate the coexistence of mixed numerologies, the system bandwidth is divided into several bandwidth parts (BWPs) of arbitrary width, each with distinct numerology optimized for a particular service. For this reason, the two terms “BWP” and “numerology” are used interchangeably in the thesis whenever no ambiguity arises. Without loss of generality, one user per numerology is assumed. A communication system with total bandwidth B is considered to support a family of M numerologies, Snum = {1, 2, ..., M }, which are related to each other via scaling, i.e., ν (i) ∆f (i) = , ∆f (j) ν (j) (i) Ncr (j) Ncr = ν (j) , ν (i) (3.1) (i) where ∆f (i) and Ncr denote subcarrier spacing and number of cyclic redundancy (CR) in samples in the i-th numerology. ν (i) ∈ N is a scaling factor. The scaling factors are chosen such that a subcarrier spacing being an integer that can be divided by all smaller ones, i.e., (i) ν (i) = 2µ , 22 ∀i ∈ Snum , (3.2) 23 where µ(i) ∈ {0, 1, 2, · · · }. By doing so, the symbol length in a numerology is always integral multiple of that in bigger numerologies. Assume that signals of different numerologies are processed in the same approach in the transceiver structure as depicted in Fig. 3.1 (b). To conserve space, we only provide the detailed description of one numerology (the i-th numerology), which is also assumed as the numerology of interest while other signals serve as interference (i) sources. Assume M (i) + 2G(i) consecutive subcarriers in the range of M̃(i) = {M0 − (i) (i) G(i) , M0 − G(i) + 1, ..., M0 + M (i) + G(i) − 1} are assigned to the i-th BWP with a subcarrier spacing ∆f (i) and a corresponding waveform shaping filter denoted as vector u(i) in the time domain at the transmitter. G(i) < M (i) /2 subcarriers from each side of the i-th BWP are reserved as guard subcarriers which do not carry any data symbols. For easy track, we denote the M (i) data bearing subcarriers in BWP(i) as (i) (i) (i) M(i) = {M0 , M0 + 1, ..., M0 + M (i) − 1}. To avoid fractional subcarrier shifts, i.e., subcarrier frequencies should coincide with the natural grid of their numerology, guard bands are allocated inside BWPs [77], as illustrated in Fig. 3.1 (a) and (c). The guard band width between BWP(i) and BWP(i+1) is G(i) ∆f (i) + G(i+1) ∆f (i+1) . Representing K consecutive OFDM symbols in the i-th numerology in a sub-frame as an M (i) K dimensional vector yields T T (i) T (i) T (i) (i) d = d0 , d1 , · · · , dK−1 , (3.3) iT h (i) (i) (i) (i) (i) where the k-th OFDM symbol dk = dk,0 , dk,1 , · · · , dk,M (i) −1 ∈ CM ×1 with the (i) individual element dk,m corresponding to the data symbol carried on the m-th subcarrier in the k-th OFDM window of the i-th numerology. The data are assumed to symbols H (i) 2 (i) (i) = σs I, be independent and identically distributed (i.i.d.) with E dk dk (i) where (σs )2 is the average transmission power of quadrature amplitude modulation (QAM) symbols for the i-th numerology. The baseband transmitter and receiver procedures are discussed in the sequel. 24 Time Tx numerology 1 numerology i d(1) Subcarrier(1) Mapping .. . IFFT & (1) Add CP Filter u(1) d(i) Subcarrier(i) Mapping IFFT & (i) Add CP Filter u(i) IFFT &(M ) Add CP Filter u(M ) .. . numerology M d (M ) Subcarrier(M ) Mapping Rx for numerology i RB guard band Rmv. CP(i) & FFT Filter v(i) Eq. & Detector d̂(i) Frequency (b) (a) … BWP(i−1) BWP(i) BWP(i+1) (M (i) + 2G(i) )∆f (i) (M (i−1) + 2G(i−1) )∆f (i−1) filter spectral envelope … (M (i+1) + 2G(i+1) )∆f (i+1) data SCs G(i) ∆f (i) (c) guard SCs guard SCs M (i) ∆f (i) Frequency Figure 3.1: System model of OFDM/F-OFDM in the presence of mixed numerologies with u(i) = 1 for OFDM. (a) An exemplary illustration of regular resource grid. (b) Downlink transceiver structure. (c) An exemplary illustration of multiplexing of mixed numerologies in the frequency domain. (RB stands for resource block, SC refers to subcarrier) 25 3.1. Transmitter Baseband Processing 3.1 Transmitter Baseband Processing The transmitter baseband processing involves operations prior to the transmission, which includes OFDM modulation and filtering on each BWP and mixing signals from all the BWPs. 3.1.1 OFDM Modulation With respect to the i-th numerology, an N (i) -point (N (i) = B ∆f (i) ≥ M (i) ) IFFT oper- ation is performed on a per OFDM symbol basis, followed by the insertion of a cyclic (i) redundancy (CR) of length Ncr for eliminating/mitigating inter-symbol interference (i) (ISI). The CR is made up of two parts, cyclic prefix (CP) of length Ncp and cyclic (i) (i) (i) (i) suffix (CS) of length Ncs (Ncr = Ncp + Ncs ), with the former being adopted for combating forward ISI, and the latter for alleviating the backward ISI (b-ISI) . To be compatible with CP-OFDM, CR can be implemented as an extended CP which incorporates CP and CS at the transmitter, and the FFT window is moved backward by the length of CS at the receiver. The k-th OFDM symbol can be expressed in the form of matrix multiplication as (i) (i) ×1 (i) (i) (i) L xk = ρ(i) cr Tcr F dk ∈ C , (3.4) (i) where xk is a vector of dimension L(i) = N (i) + Ncr . F(i) is an N (i) × M (i) submatrix of N (i) -point IFFT matrix defined by its element on the n-th row and m-th column as ! (i) p H j2πn m+M0 , and it is unitary such that F(i) F(i) = (F (i) )n,m = 1/N (i) exp N (i) (i) IM (i) . Tcr = (i) (i) T Icp T T (i) , IN (i) , Ics is an L(i) × N (i) dimensional CR insertion (i) (i) (i) matrix, with Icp and Ics containing the last Ncp and the first Ncs rows of the identity p (i) matrix IN (i) , respectively. ρcr = N (i) /L(i) is the power normalization factor. 3.1.2 Transmitter Filtering The OFDM symbols are post-processed by an appropriately designed spectrum shaping filter. The actual transmitted F-OFDM signal of the i-th numerology is finally obtained 26 3.1. Transmitter Baseband Processing ∗ = forward spreading backward spreading (i) Nu 2 U(i:l) U(i:o) backward spreading terms U(i:u) L (i) (i) Nu 2 forward spreading terms L Figure 3.2: Matrix shapes of filter forward/backward spreading as s(i) = x(i) ∗ u(i) , (3.5) h iT (i) (i) (i) where x(i) = x0 , x1 , · · · , xK−1 , and h iT (i) (i) (i) u(i) = u0 , u1 , · · · , u (i) (3.6) Nu (i) is a length (Nu + 1) vector representing the impulse response of the transmitter filter with u(i) = [1] corresponding to the conventional OFDM. It is placed into the center (i) of the assigned BWP and occupies its entire bandwidth. Filter length Nu is chosen to be an even number for ensuring the time domain symmetry, and less than or equal to (i) half of the FFT size, i.e., Nu ≤ N (i) /2, to restrict signal spreading within one OFDM symbol duration. Fig. 3.2 shows how the Intra-NI is introduced by filtering. As depicted in the figure, fISI/b-ISI are inflicted by the forward/backward spreading of the previous/next OFDM symbols to the current FFT window . The ICI of the current OFDM symbol comes from the energy loss due to its bi-directional spreading. To facilitate interference analysis, we derive the equivalent matrix form of a linear filtering process. The L(i) received samples (i) relative to the k-th OFDM symbol are grouped into the vector sk , thus obtaining (i) (i) (i) (i) sk = U(i:u) xk−1 + U(i:m) xk + U(i:l) xk+1 , (3.7) 27 3.1. Transmitter Baseband Processing where the L(i) × L(i) forward spreading matrix U(i:u) is a strictly upper triangular matrix with its (r, c)-th element being defined as (i) u(i)(i) , c ≥ r + L(i) − N2u Nu (i:u) +L(i) +r−c Ur,c = 2 (i) 0, c < r + L(i) − N2u 0 ≤ r, c ≤ L(i) − 1 . (3.8) It is of the form U(i:u) 0 . .. = 0 . . . 0 (i) ··· u (i) Nu .. .. ··· u .. . .. .. . (i) u (i) Nu . .. . ··· 0 .. 0 (i) Nu +1 2 . . . ··· (i) . (3.9) T (i) (i) The toeplitz matrix U(i:m) is specified by its first column u (i) Nu 2 ,··· ,u (i) Nu ,0 (i) N 1× L(i) − u −1 2 T (i) (i) and first row u (i) , · · · , u0 , 0 Nu 2 . It is a matrix in which the only (i) N 1× L(i) − u −1 2 (i) Nu 2 nonzero entries are on main diagonal and the first diagonals above and below. It is of the form U (i:m) (i) u Nu(i) 2 .. . (i) u (i) Nu = 0 . . . 0 (i) · · · u0 0 ··· .. .. .. . .. . . .. . . .. .. . .. . .. ··· 0 . .. . (i) u (i) Nu . ··· 0 .. . 0 . (i) u0 .. . (i) u (i) (3.10) Nu 2 The backward spreading matrix U(i:l) is a strictly lower triangular matrix with its (r, c)-th element being defined as (i) , r ≥ c + L(i) − u (i) N u (i) (i:l) −1−L +r−c Ur,c = 2 0, r < c + L(i) − (i) Nu 2 (i) Nu 2 +1 +1 0 ≤ r, c ≤ L(i) − 1 . (3.11) 28 3.1. Transmitter Baseband Processing Time (Symbol) Frequency µ(i) − 1 (i) 2T (i) ∆f /2 T (i) µ(i) (i) ∆f T (i) 2 (i) 2∆f µ(i) + 1 Numerology Figure 3.3: An example of symbol overlap among different numerologies. It is of the form U(i:l) 0 .. . (i) = u0 .. . (i) u (i) Nu 2 3.1.3 −1 ··· 0 . . .. . . .. . 0 . . .. .. . . . . · · · u0 · · · 0 ··· .. . 0 (3.12) Multiplexing of Mixed Numerologies Signals from all numerologies are added prior to the transmission. Assume that multiplexing different numerologies is performed within a subframe. As shown in Fig. 3.3, numerologies have different symbol durations. OFDM symbols of a numerology with smaller subcarrier spacing have a longer duration, leaving OFDM symbols overlapped in a more complicated way compared to a single numerology scenario. Taking the symbol duration of the i-th numerology as a reference, we can divide the set of numerologies 29 3.1. Transmitter Baseband Processing i+ Snum into two subsets Si− num and Snum with a smaller and greater subcarrier spacing than the i-th numerology, respectively. They are defined as (i−) Snum = {j : µ(j) < µ(i) ; i, j ∈ Snum } (j) and S(i+) > µ(i) ; i, j ∈ Snum }. (3.13) num = {j : µ Assume perfect synchronization for all numerologies, we have the following propositions for multiplexing two numerologies (proof can be found in Appendix A): (i−) Proposition 3.1. The symbol duration of the j-th numerology (j ∈ Snum ) is ν (i) ν (j) times long as that of the i-th numerology. If we divide each OFDM symbol of the j (i) (i) th numerology into νν(j) symbol parts (SPs) of same length, then the k mod νν(j) -th l (j) m (i←j) SP of the k νν (i) -th OFDM symbol, denoted as sk , exactly overlaps with the k-th (i←j) OFDM symbol of the i-th numerology. The SP sk can be obtained by slicing the (i) corresponding part from the k mod νν(j) -th OFDM symbol of the j-th numerology, (j) sl (j) k ν (i) m, as ν (i←j) sk (i←j) (j) = Ck s (j) dk ν (i) e j ∈ S(i−) num , , ν where (i←j) Ck is a (i←j) Ck L(i) = × L(j) (3.14) dimensional slicing matrix defined as 0 (i) L × k mod , I (i) , 0 (i) (j) ν (i) L(i) L L × L − k mod (j) ν ν (i) +1 L(i) (j) . ν Proposition 3.2. The symbol duration of the i-th numerology is ν (j) ν (i) times as long as (i+) that of the j-th numerology (j ∈ Snum ). The k-th symbol of the i-th numerology overlaps with ν (j) ν (i) symbols of the j-th numerology in the range of (i←j) Grouping these symbols into an L(i) × 1 vector sk !T !T (i←j) sk (j) = s kν (j) ν (i) (j) , s kν (j) ν (i) kν (j) kν (j) , ν (i) ν (i) , · · · , s (k+1)ν (j) ν (i) (j) − 1. yields !T T (j) +1 + 1, · · · , (k+1)ν ν (i) −1 , (i+) j ∈ Snum . (3.15) By multiplexing signals from all numerologies, the total transmitted signal with respect to the k-th OFDM symbol of the i-th numerology can be written as s<i> = sk + s̃k , k (i) (i) (3.16) (i) (i←j) (3.17) where s̃k = X sk j∈S\{i} is the multiplexing signal from numerologies other than the i-th numerology. 30 3.2. Passing Signal Through the Channel 3.2 Passing Signal Through the Channel (i) An (Nch + 1)-tap channel between the transmitter and the receiver of the i-th numerology is assumed to have an impulse response h i (i) (i) (i) T h(i) = h0 , h1 , ..., hNch . (3.18) After passing through the above channel followed by adding an additive white Gaussian noise (AWGN), the L(i) received samples corresponding to the k-th OFDM symbol of the i-th numerology can be written as (i) (i:m) <i> r<i> = H(i:u) s<i> sk + wk , k k−1 + H (3.19) (i) ×1 where H(i:m) is a Toeplitz matrix with [(h(i) )T , 01×(L−Nch −1) ]T ∈ CL column and (i) [h0 , 01×(L−1) ]T being its first L(i) ×1 ∈C H(i:m) being its first row. It is of the form (i) h0 0 ··· ··· 0 .. .. .. .. . . . . .. .. .. h(i) . . . = (i) . N ch . .. .. .. . . 0 (i) (i) 0 · · · h (i) · · · h0 (3.20) Nch The channel spreading matrix H(i:u) is a strictly upper triangular matrix with its (r, c)th element being defined as h(i)(i) , L +r−c (i:u) Hr,c = 0, (i) c ≥ r + L(i) − Nch otherwise 0 ≤ r, c ≤ L(i) − 1 . (3.21) It is of the form H(i:u) 0 . . . = 0 .. . 0 (i) ··· h .. (i) Nch . .. .. . ··· 0 . ··· (i) h1 .. . . (i) . h (i) Nch .. .. . . ··· 0 .. (3.22) 31 3.3. Receiver Baseband Processing (i) wk is the addictive white Gaussian noise (AWGN) vector with each element having a zero mean and variance σn2 . By replacing s<i> in (3.19) with its expression in (3.16), we can decompose the vector k r<i> as k (i) (i) (i) r<i> = rk + r̃k + wk , k (3.23) where (i) (i) (i) rk = H(i:u) sk−1 + H(i:m) sk (3.24) (i) (3.25) (i) (i) r̃k = H(i:u) s̃k−1 + H(i:m) s̃k . refer to the contributions to r<i> from the signal of i-th numerology and the others, k respectively. 3.3 3.3.1 Receiver Baseband Processing Filtering at the Receiver The primary purpose of the filtering at the receiver is to reject the signal contribution from other numerologies. To comply with the 3GPP’s requirement on transparent filtering, we assume that the transmitter filtering parameter is not known at the receiver and define an independent receiver filter of length Nv as h i (i) (i) (i) T v(i) = v0 , v1 , · · · , vNv . (3.26) When no filtering is performed at the receive, we have v(i) = 1. The L(i) samples corresponding to the k-th OFDM symbol passing through the receiver filter are grouped into the vector z<i> , obtaining k (i:m) <i> z<i> = V(i:u) r<i> rk + V(i:l) r<i> k k−1 + V k+1 , (3.27) where the L(i) × L(i) forward spreading matrix V(i:u) is a strictly upper triangular matrix with its (r, c)-th element being defined as (i) (i) , c ≥ r + L(i) − N2v v (i) Nv (i:u) +L(i) +r−c Vr,c = 2 (i) 0, c < r + L(i) − Nv 2 0 ≤ r, c ≤ L(i) − 1 . (3.28) 32 3.3. Receiver Baseband Processing The toeplitz matrix V(i:m) is specified by its first column T v (i)(i) , · · · , v (i)(i) , 0 (i) N 1× L(i) − v2 −1 Nv Nv 2 and first row T v (i)(i) , · · · , v0(i) , 0 Nv 2 (i) N 1× L(i) − v2 −1 . It is a matrix in which the only nonzero entries are on main diagonal and the first (i) Nv 2 diagonals above and below. The backward spreading matrix V(i:l) is a strictly lower triangular matrix with its (r, c)-th element being defined as v (i)(i) , r ≥ c + L(i) − Nv (i) (i:l) −1−L +r−c vr,c = 2 0, r < c + L(i) − (i) Nv 2 (i) Nv 2 +1 +1 0 ≤ r, c ≤ L(i) − 1 . (3.29) An ideal low-pass filter rejects all signal energy above a designated cut-off frequency. However, it is practically impossible as the required impulse response would be infinitely long. Practical finite-length filters inevitably result in residual signals above the cut-off (i) frequency. Thus, zk comprises signal components not only from the i-th numerology, but also from others. Denote zk as the sum of three terms, i.e., (i) (i) (i) z<i> = zk + z̃k + w̃k , k (3.30) where (i) (i) (i) (i) (3.31) (i) (i) (i) (i) (3.32) zk = V(i:u) rk−1 + V(i:m) rk + V(i:l) rk+1 , z̃k = V(i:u) r̃k−1 + V(i:m) r̃k + V(i:l) r̃k+1 , correspond to the signal components from the i-th numerology and the residual interference signal to the i-th numerology, respectively. Their expression in terms of OFDM symbols before any filtering operation are derived in Appendix B and C. The filtered (i) noise w̃k can be expressed as (i) (i) (i) (i) w̃k = V(i:u) wk−1 + V(i:m) wk + V(i:l) wk+1 . (3.33) 33 3.3. Receiver Baseband Processing 3.3.2 OFDM Demodulation After removing CR and applying FFT, the received symbol vector corresponding to the k-th OFDM of the i-th numerology can be obtained as H <i> R(i) , yk<i> = F(i) cr zk (3.34) (i) where the CR removal matrix is formed as Rcr = [0N (i) ×N (i) , IN (i) , 0N (i) ×N (i) ] ∈ cp BN (i) ×L(i) cs . After some basic algebraic manipulations based on equations from (3.4) to (3.34), the received signal yk<i> can be decomposed as yk<i> = (i) yk,des | {z } (i) + desired signal yk,intra | {z } (i) + Intra-NI signal yk,inter | {z } Inter-NI signal (i) + ŵk . |{z} (3.35) noise The desired signal can be expressed as (i) (i) (i) yk,des = Ψdes dk , (3.36) (i) where the M (i) × M (i) dimensional matrix Ψdes , which is the equivalent channel matrix to the desired signal, can be expressed as H (i) (i) (i) (i) (i) Ψdes = ρ(i) F R(i) cr cr Θ Tcr F , (3.37) with Θ(i) = V(i:u) H(i:m) U(i:l) +V(i:m) H(i:u) U(i:l) +V(i:m) H(i:m) U(i:m) +V(i:l) H(i:m) U(i:u) . The derivation can be found in Appendix B. The interferences, Intra-NI and Inter-NI in (3.35) will be discussed in the following two chapters: Chapter 4 and Chapter 5. The 3.3.3 Equalization and Detection The well-channelized signal (3.35) can be equalized using the classic equalization methods with a trade-off between complexity and performance. The most simplest method is the one-tap equalization using diagonal matrix Ψdes , obtained as (i) (i) d̂k = Ψ−1 des yk , which can be performed independently on each subcarrier. (3.38) 3.4. Summary 3.4 34 Summary In this chapter, we established a generic analytical framework for OFDM/ F-OFDM systems to address the 5G NR numerology coexistence issue. Specifically, numerologies and frame structure were defined, and a transceiver structure was proposed complying with the 3GPP’s requirement on transparent waveform processing in which the receiver has no knowledge of filtering parameters employed at the transmitter. In the model, two prepositions were introduced to formulate the process of multiplexing mixed numerologies. With the derived model, all filtering operation was expressed in matrix form, and the interference signal was divided into Inter-NI and Intra-NI, which will be investigated in detail in the following two chapters. More advanced equalization methods can be deployed to enhance system performance, which will be discussed in more detail in Section 4.2.5. Chapter 4 Intra-Numerology Interference and Filter Selectivity Analysis As mentioned earlier, the upcoming 5G and beyond wireless networks will face challenges arising from use cases/services with extremely divergent QoS requirements. The F-OFDM, as a representative subband filtered waveform, can be employed as a framework to perform radio partition with programmable numerologies tailored/optimized for each service to meet its unique QoS requirements. However, the additional filtering operation on top of the legacy OFDM will affect the performance in various aspects. In this chapter, we will investigate three consequences inflicted within a BWP in F-OFDM systems: filtered-noise, Intra-NI and FFRS. Section 4.1 describes the noise distribution in F-OFDM systems. In Section 4.2, the exact-form expression for the Intra-NI is derived as a functions of subband parameters and a low-complexity block wise parallel interference cancellation (BwPIC) algorithm is proposed accordingly. The Intra-NI-free and nearly-free conditions of F-OFDM systems are also discussed and defined as a guidance to select the optimal cyclic redundancy length. It is worth mentioning that all the considerations of this chapter (noise, Intra-NI and FFRS) within one numerology are independent to those of the other numerologies. The interference between different numerologies, i.e., Inter-NI, has no impact on them. 35 36 4.1. Noise Distribution in F-OFDM Systems Therefore, for simplicity, we assume that the Inter-NI is well eliminated through the methods discussed in Chapter 5, and focus on the issues occurring within a BWP, e.g., the BWP(i) (∀i ∈ Snum ). To this end, the system model described in Chapter 3 is effectively reduced to a single-numerology system. 4.1 Noise Distribution in F-OFDM Systems In comparison to CP-OFDM, the noise in F-OFDM passes an added Rx filter. Intuitively, the distribution of noise in the frequency domain depends on the distribution property of the filter it passes. The analytical expression of the noise and its power distribution are derived in the following. According to (3.33), we can see that the M (i) × 1 complex noise vector added on the received data symbols of the k-th F-OFDM symbol can be reformulated as H (i) (i:u) ŵk = F(i) R(i) wk−1 + V(i:m) wk + V(i:l) wk+1 cr V H (i:[u,m,l]) (i) = F(i) R(i) w[k−1,k,k+1] , cr V (4.1) where the L(i) × 3L(i) matrix V(i:[u,m,l]) = V(i:u) , V(i:m) , V(i:l) . The 3L(i) × 1 vech iT (i) (i) (i) (i) tor w[k−1,k,k+1] = wk−1 T , wk T , wk+1 T is a standard Gaussian random vector with (i) (i) w[k−1,k,k+1] ∼ N(03L(i) ×1 , σn2 I3L(i) ). The covariance matrix of vector ŵk can be computed as C(i) w H H H (i) (i) H (i) (i) (i:[u,m,l]) Rcr V V(i:[u,m,l]) Rcr F(i) σn2 . = E ŵk ŵk = F(i) (4.2) (i) (i) (i) As a result, ŵk is a length M (i) Gaussian random vector with ŵk ∼ N(0M (i) ×1 , ηnoise ), (i) (i) and the column vector ηnoise = diag(Cw ) ∈ RM (i) ×1 represents average noise power on each subcarrier of the subband. 4.2 Intra-Numerology Interference Analysis 5G NR [18] recommends to use spectral confinement techniques, such as filtering or windowing, to improve spectrum localization among different BWPs, as long as the 37 4.2. Intra-Numerology Interference Analysis backward spreading data (k − 1)th symbol forward spreading k th symbol (k + 1)th symbol ICI ICI forward ISI backward ISI Figure 4.1: Illustration of intra-numerology interference techniques employed at the Tx are transparent to the Rx. However, it is not possible to have a better frequency localization without compromising in time localization according to the Heisenberg’s principle [81]. When filtering applies on top of CP-OFDM, the OFDM symbol spreads in the time domain as spectrum localization improves. In the case when the dispersion exceeds the coverage of CR, the orthogonality among subcarriers in the same numerology is destroyed and Intra-NI occurs. As illustrated in Fig. 4.1, signal spreads bi-directionally and inflicts interference between adjacent OFDM symbols. The interference signal to the k-th OFDM symbol can be divided into three components: f-ISI, b-ISI and ICI. The f-ISI is inflicted by the forward spreading of the (k − 1)-th OFDM symbols to the k-th FFT window, while the b-ISI is caused by the backward spreading of the (k + 1)-th OFDM symbols to the k-th FFT window. The ICI comes from the energy loss due to its bi-directional spreading. 4.2.1 The Expression of the Intra-NI Signal It is derived in Appendix B that the signal vector obtained after filtering at the receiver is (i) (i) (i) (i) (i) (i) xk−1 + Θ(i) xk + Θnext xk+1 , zk = Θpre (4.3) 38 4.2. Intra-Numerology Interference Analysis where (i:u) (i:m) (i:m) Θ(i) H U + V(i:m) H(i:u) U(i:m) + V(i:m) H(i:m) U(i:u) , pre = V Θ(i) = V(i:u) H(i:m) U(i:l) + V(i:m) H(i:u) U(i:l) + V(i:m) H(i:m) U(i:m) + V(i:l) H(i:m) U(i:u) , (i) Θnext = V(i:m) H(i:m) U(i:l) + V(i:l) H(i:u) U(i:l) + V(i:l) H(i:m) U(i:m) . (4.4) (i) By replacing xk in (4.3) with its expression in (3.4), we have (i) (i) (i) (i) (i) (i) yk = Ψf-ISI dk−1 + Ψ(i) dk + Ψb-ISI dk+1 , (4.5) where (i) H (i) (i) (i) (i) Ψ(i) = ρ(i) cr (F ) Rcr Θ Tcr F , (i) (i) H (i) (i) (i) (i) Ψf-ISI = ρ(i) cr (F ) Rcr Θpre Tcr F , (i) (i) (i) H (i) (i) (i) Ψb-ISI = ρ(i) cr (F ) Rcr Θnext Tcr F . (i) (4.6) (i) By defining an M (i) × M (i) diagonal matrix Ψdes = diag(diag(Ψ(i) )), and ΨICI = (i) Ψ(i) − Ψdes accordingly, we obtain (i) (i) (i) (i) yk = Ψdes dk + yk,intra , (4.7) (i) where the Intra-NI signal yk,intra is formed as the sum of its three components as (i) (i) (i) (i) (i) (i) (i) yk,intra = ΨICI dk + Ψf-ISI dk−1 + Ψb-ISI dk+1 . | {z } | {z } | {z } ICI signal 4.2.2 f-ISI signal (4.8) b-ISI signal Channel Diagonalization and Intra-NI-free Systems Optimal performance can be achieved by the simple one-tap equalization in an interferencefree system, in which the channel is completely diagonalized. The condition to ensure an IntraNI-free F-OFDM system will be derived in the sequel. (i) The matrix Θpre can be easily proved to be a strictly upper triangular matrix by following the same approach used in Appendix D, of which all non-zero elements are on the top Nu + Nch rows. When the length of CP is chosen as (i) (i) Ncp ≥ Nu(i) + Nch , (4.9) 39 4.2. Intra-Numerology Interference Analysis (i) the resulting CP removal matrix Rcr can sufficiently remove all non-zero elements (i) (i) of Θpre . This leads to a zero f-ISI inducing matrix Ψf-ISI . Similarly, the following condition (i) Ncs ≥ Nv(i) (4.10) ensures a zero b-ISI inducing matrix. With (4.9) and (4.10) being met, the insertion of CR converts the linear convolution q(i) ∗ h(i) ∗ p(i) ∗ x(i) into a circular convolution q(i) ~ h(i) ~ p(i) ~ x(i) . The symbol (i) vector yk in (4.7) can be expressed as (i) (i) yk = Λ(i) dk , (4.11) where the diagonalized channel matrix Λ(i) = Q(i) H(i) P(i) with Q(i) = diag(q¨(i) ), H(i) = diag(h¨(i) ), P(i) = diag(p¨(i) ). ẍ (x ∈ {p, q, h}) refers to DFT of x on the correspond(i) ing subcarriers. The equation (4.11) implies that Ψ(i) = Ψdes = Λ(i) and a zero ICI (i) inducing matrix ΨICI . As a consequence, the CR length specified in (4.9) and (4.10) is the condition to achieve Intra-NI-free F-OFDM systems. Based on the discussion above, we form the following Proposition: Proposition 4.1. Consider a multi-numerology system depicted in Fig. 3.1 with the transmitter, the channel, and the receiver filter being defined in (3.6),(3.18),(3.26). It is an Intra-NI-free system under the condition of perfect synchronization at the receiver if (i) (i) Ncp ≥ Nu(i) + Nch and (i) Ncs ≥ Nv(i) , ∀i ∈ Snum (4.12) and the received data symbol can be expressed as in (4.11). 4.2.3 The Analytical Expression of the Intra-NI Power When the Intra-NI-free condition is violated, the system will be not strictly orthogonal within a numerology. In the time domain, the signal from adjacent OFDM symbols 40 4.2. Intra-Numerology Interference Analysis spreads into the FFT window of the OFDM symbol of interest, which produces the forward and backward ISI. In the frequency domain, the extended part of the interested OFDM symbol falls out the range of CP/CS and leads to the energy lost. As a result, the matrix Ψ is no longer diagonal, causing the inter-carrier interference. The instantaneous power of the desired signal and interference signal (f-ISI, b-ISI and ICI) on all subcarriers in one transmission block can be grouped as a (M (i) − 2G(i) ) × 1 vector, obtained as (i) 2 ydes = diag (i) (i) Ψdes dk (i) H dk (i) H Ψdes , (i) (i) (i) H (i) H = diag ΨICI dk dk ΨICI , H H (i) 2 (i) (i) (i) (i) yf-ISI = diag Ψf-ISI dk−1 dk−1 Ψf-ISI , H H 2 (i) (i) (i) (i) (i) yb-ISI = diag Ψb-ISI dk+1 yk+1 Ψb-ISI , (i) 2 yICI (i) yIntra 2 (i) 2 = yICI (i) + yf-ISI 2 (i) (4.13) 2 + yb-ISI . The expression of instantaneous power for Intra-NI can be helpful in studying the system behavior in fading environment for utilizing its diversity, For example, maximizing system performance with respect to spectrum or power efficiency through subcarrier and power allocation. However, the primary purpose of this paper is to analyze the interference induced by filtering processing, other sources of distortion such as multi-path channels, are omitted by forcing the channel matrix to be identity (H(i,u) = 0, H(i,u) = I) (i) in the expression of Ψx , x ∈ {des, ICI, f-ISI, b-ISI} in (4.6). By doing so, the power of Intra-NI is reduced into a function of the bandwidth of the BWP, CR length, and filter parameters as ȳx (M (i) (i) , Ncr , u(i) , v(i) ) 2 (i) (i) (i) H (i) H = E diag Ψx dk (dk ) (Ψx ) H = diag Ψx(i) (Ψ(i) ) (σs(i) )2 x x ∈ {des, ICI, f-ISI, b-ISI}. (4.14) 41 4.2. Intra-Numerology Interference Analysis 4.2.4 Intra-NI Mitigation: A Practical Approach for Choosing CR Length Proposition 4.1 implies that IntraNI-free systems can be achieved by adding a sufficient number of redundant samples. The implementation of CR, although elegant and simple, is not entirely free. It comes with a bandwidth and power penalty. Since Ncr redundant samples are transmitted, the actual bandwidth for F-OFDM increases from B to Ncr +N B. N Similarly, an additional Ncr samples must be counted against the transmit power budget resulting in a power loss of 10 log10 Ncr +N N dB. For an F-OFDM system with stringent frequency localization requirement, the filter length is normally chosen up to half of the symbol duration, making the satisfaction of interference-free condition in (4.12) unaffordable with respect to the power and bandwidth loss. On the one hand, interference-free systems are preferred due to the benefit to the computational complexity reduction and the signal-to-interference and noise ratio (SINR) improvement. On the other hand, a highly efficient system requires shorter overhead (CP/CS). Therefore, choosing the size of CR, as a tradeoff between the two contradicting parties, forms an optimization problem. However, it is very hard to find an optimal solution due to the multi-objective characteristic of the problem. A sub-optimal attempt in [71] is suggested by setting the length of overhead to the width of the main lobe, due to the fact that the main lobe of a sinc filter carries most of its energy. However, it neglects the fact that filters of different bandwidth vary in energies captured by the main lobes. Wider subband filters have less energy enclosed in the main lobes than that of narrower subband filters. We take this into consideration and propose a BWP width-dependent approach for choosing the CR length. For a length (K +1) filter defined as f = [f0 , f1 · · · , fK ]T , occupying a subband of width equivalent to M subcarriers in a channel of N subcarriers, the energy rate consisting in the k middle samples is defined as P K2 + k2 ζ(f , k) = f f∗ m= K − k2 m m 2 PK ∗ m=0 fm fm (4.15) Then, the number of overhead can be chosen to ensure that the minimum energy captured by the CR greater than a pre-defined value. We define the following proposition: 42 4.2. Intra-Numerology Interference Analysis Proposition 4.2. Consider a multi-numerology system depicted in Fig. 3.1 with the transmitter filter, the channel and the receiver filter defined in (3.6), (3.18), and (3.26). It is considered as a nearly Intra-NI-free system under the condition of perfect synchronization at the receiver if (i) (i) Ncp ≥ Ku(i) + Nch (i) and (i) Ncs ≥ Ku(i) , ∀i ∈ Snum (4.16) (i) where Ku and Kv are selected to satisfy arg min ζ(u(i) , Ku(i) ) ≥ α (i) Ku and arg min ζ(u(i) , Ku(i) ) ≥ α (4.17) (i) Ku with α being a pre-defined value, e.g., α = 0.99. The received data symbol can then be approximated by (4.11) with trivial interference small enough to be ignored, and the effective channel is nearly diagonal1 . When the condition in (4.16) is met, it can be seen from the numerical results in Fig. 4.7 that the power of effective interference, i.e., the maximum total power of ICI, forward ISI, backward ISI, reduces to a level very close to -30 dB. Eq. (4.16) is named as a nearly Intra-NI-free condition of F-OFDM systems. The solutions to the optimization problems in (26) can be obtained using a linear search in a sorted list (1,2,...,K). In particular, the linear search sequentially checks each element of the list and evaluates ζ in (24) until it finds the first CP length that satisfies the specified condition in (26). The denominator of ζ is a constant value (only calculated once), and the numerator is an accumulated term. Therefore, the calculation of ζ at each iteration comprises an addition, a multiplication, and a division, i.e., the computation complexity of each iteration is constant. In the worst cases, it makes K comparisons. However, the optimal CP length can be found close to the width of the first main lob due to the energy distribution nature of the filter. The main lob of a filter has bN/M c samples, therefore the complexity of the search algorithm is O(N/M ). 1 How close the effective channel to a diagonal matrix can be measured quantitatively by Frobenius norm of the matrix Ψe . The smaller the ||Ψe ||F is, the closer the effective channel to a diagonal matrix. ||Ψe ||F = 0 indicates a perfect diagonal matrix. 43 4.2. Intra-Numerology Interference Analysis 4.2.5 An Alternative for Intra-NI Mitigation: Frequency Domain Equalization Linear equalizers Two representative equalizers, i.e., zero-forcing (ZF) and minimum mean square error (MMSE), apply an equalization matrix to the current symbol to reverse the effective channel effect. Considering that the received signal of the k-th F-OFDM symbol can be expressed as (i) (i) yk = Ψdes dk + interference terms + ŵk , (4.18) we define ZF and MMSE equalizers in f-OFDM systems as −1 (i) H (i) (i) H zf Eeq = Ψdes Ψdes Ψdes , Emmse eq = (i) H Ψdes (i) H Ψdes (i) Ψdes + (i) ηnoise −1 . (4.19) Non-linear equalizers A non-linear equalizer may improve the performance relative to linear receivers by employing interference cancellation (IC) techniques. Conventional successive interference cancellation (SIC) algorithms come with significantly high computational complexity due to their high cancellation granularity. We propose a novel interference cancellation algorithm customized for F-OFDM systems, namely, blockwise parallel interference cancellation (BwPIC), which performs cancellation once only for all F-OFDM symbols in a data block per iteration. The algorithm comes with lower complexity than SIC because the cancellation is only carried out on a block basis. The algorithm cancels the Intra-NI of all F-OFDM symbols in one block in parallel. The detail is shown in Algorithm 1. It involves a sequence of interference cancellation/equalization/slicing1 operations. At each iteration of the outer loop, a vector h iT d̃ = d̃0 , · · · , d̃K−1 is updated, and d̃pre /d̃next are derived accordingly. Then the interference corresponding to a whole block of F-OFDM symbols is canceled simultaneously according to (4.7) as (i) (i) (i) (i) ŷ(i) = y(i) − ΨICI d̃(i) − Ψf-ISI d̃(i) pre − Ψb-ISI d̃next . 1 (4.20) slicing refers to QAM symbol slicing which detects a QAM symbol from a distorted signal. 44 4.3. Filter Selectivity Analysis and Discussion However, the equalization and slicing are performed on a single F-OFDM symbol basis at each iteration of the inner loop. One-tap equalization is adopted in the algorithm for reducing computational complexity. The slicing operation approximates an equalized symbol to its nearest QAM point in the constellation. Algorithm 1 Block-wise Parallel Interference Cancellation (BwPIC) (i) 1: Inputs: y(i) , ΨICI , Ψf-ISI , Ψb-ISI , I 2: output: d̃(i) 3: Initialization: d̃(i) = 0KM(i) ×1 4: for i = 1; i <= I; i++ do 5: 6: h iT (i) (i) (i) d̃pre = 0M (i) ×1 , d̃1 T , · · · , d̃K−1 T , h iT (i) (i) T d̃next = d̃T , 0M(i) ×1 , (i) 2 , · · · , d̃K (i) (i) (i) (i) (i) 7: ŷ(i) = y(i) − ΨICI d̃(i) − Ψf-ISI d̃pre − Ψb-ISI d̃next (interference canceling) 8: for k = 1; k <= K; k++ do 9: 10: 11: (i) d̂k = Eŷ(i) (i) (i) d̃k = Q(d̂k ) (one-tap equalization) (Slicing) end for 12: end for (i) 13: return d̃k 4.3 Filter Selectivity Analysis and Discussion In this section, we continue to investigate another issue within a numerology, FFRS, induced by the filtering operations in F-OFDM systems, which may cause system performance degradation. An ideal low-pass filter is the one that completely eliminates all frequencies above a designated cutoff frequency, while leaving those below unchanged. Its frequency response is a rectangular function, as illustrated in red dotted line in Fig. 4.2. However, it is practically not possible to implement an ideal lowpass filter since the required impulse response is infinitely long. Practical filters have finite length, which inevitably leads to a nontrivial transition band between a passband and a stopband, as illustrated in the solid blue line in Fig. 4.2. 45 4.3. Filter Selectivity Analysis and Discussion Filter selectivity refers to the uneven weights of frequency response in a transition band. It reduces the power of signals on the corresponding subcarriers which become undesirable for carrying data. As a result, the bandwidth efficiency is reduced. The bandwidth loss can be computed as bandwidth loss = Nt /M, where Nt is the number of subcarriers accommodated in the transition band, and M is the total number of subcarriers in the BWP. This implies that the bandwidth loss increases linearly as the width of the BWP decreases. Although the loss for medium/wide BWP is not significant, for an extremely narrow BWP with only one physical resource block (12 subcarriers), it reaches 33% when 4 subcarriers reside in the transition band. In terms of 5G mMTC service, aiming to provide a massive number of connections, the system band is expected to be divided into many narrow BWPs. The bandwidth loss is severe in this case, which motivates us to consider exploiting these edge subcarriers to save the bandwidth. We will look into the issue in single antenna and multi-antenna systems respectively in the following subsections. 4.3.1 Filter Selectivity in Single Antenna Systems We shall use the same system model described in Chapter 3 to investigate the issue caused by FFRS in single antenna systems. For simplicity, the superscript for the considered numerology/BWP, e.g., BWP(i) , is omitted. Suppose the nearly IntraNIfree condition defined in (4.16) is satisfied. After removing the CP/CS at the receiver, the FFT output, as the demodulated received signal on subcarrier m ∈ M in the k-th F-OFDM symbol, can be approximated according to (4.11) by ym = üm v̈m ḧm dm + ŵm , (4.21) where üm , v̈m , and ḧ are complex frequency responses of the transmitter filter, the receive filter, and the channel on subcarrier m, respectively. The subscription k, representing the index of a F-OFDM symbol, is also dropped without loss of generality. Then, the average power of the desired signal on subcarrier m can be computed as 2 σ̃s,m = E üm v̈m ḧm dm (üm v̈m ḧm dm )∗ = |üm v̈m |2 σs2 . 46 4.3. Filter Selectivity Analysis and Discussion Amplitude Ideal filter Transition Passband Stopband Frequency Figure 4.2: Ideal low-pass filters versus practical filters Weight pre-equalizer 1 loss ρpre-equ filter Frequency Figure 4.3: Illustration of pre-equalizer. Due to the effect of FFRS, the amplitude of frequency response of edge subcarriers is smaller than that of the middle subcarriers. Therefore, the power of received signal on edge subcarriers has a lower value than others, i.e., the probability that edge subcarriers go to deep fade increases, and the system performance degrades on these subcarriers. To tackle this issue, we proposed a pre-equalized F-OFDM system, denoted as PFOFDM, in which, instead of directly modulating QAM complex symbols with equal powers on all subcarriers, we firstly precode the complex symbols with a weight defined as gm = ρpre-equ 1 , ∗ üm v̈m (4.22) 4.3. Filter Selectivity Analysis and Discussion 47 where ρpre-equ < 1 is power normalization factor to ensure constant power before and after precoding. The precoding inverses the uneven weight distribution of filtering and eliminate FFRS, as shown in Fig. 4.3. This enables that subcarriers in the transition band are able to carry data and avoids the bandwidth loss. However, this approach is not entirely free. It comes with a power loss of ρ2pre-equ , and the power normalization factor ρpre-equ can be calculated as s ρpre-equ = M . −1 (ü. ∗ v̈ )H (ü. ∗ v̈)−1 (4.23) It is worth mentioning that ρpre-equ is a BWP width dependent value but decreases as the width of subband grows, which indicates that narrower BWPs suffer more power loss from pre-equalization. 4.3.2 Filter Selectivity in Multi-Antenna Systems Spatial diversity, a well-known technique for combating the detrimental effects of multipath fading, can be implemented either at the receiver side or the transmitter side. A space-time block code (STBC), referred as the Alamouti code after its inventor [82], has become one of the most popular means of achieving transmit diversity due to its ease of implementation (linear both at the transmitter and the receiver) and its optimality with regards to diversity order. Originally designed for a narrow band fading channel, STBCs can be easily adapted to wideband fading channel using OFDM by utilizing adjacent subcarriers rather than consecutive symbols, referred as space-frequency block code (SFBC). In SFBC-OFDM systems, the SFBC decoder can eliminate all spatial interference under the assumption that the channel is constant over two adjacent subcarriers. This is a reasonable assumption in OFDM systems if Bc B/N , where Bc is channel coherent bandwidth, B is system bandwidth, and N is the number of subcarriers, which can be forced to be true by choosing a large enough number of subcarriers N . When SFBCs are implemented in F-OFDM systems (SFBC-F-OFDM), the FFR selectivity violates this condition and destroys the spatial orthogonality, which will be discussed in detail in the rest of the subsection. For simplicity and consistency, we use a (2 × 1) Alamouti SFBC, but the concepts apply equally to any other higher dimensional transmit and receive antennas. Fig. 4.4 48 4.3. Filter Selectivity Analysis and Discussion d d̂ Alamouti SFBC Encoder Detection Precoding g IFFT Add CP Filtering p h1 Precoding g IFFT Add CP Filtering p h2 Alamouti SFBC Deconder Rmv. CP FFT Filtering q v Figure 4.4: A block diagram of a generic filtered SFBC- OFDM system with two transmit antennas and a single receive antenna. shows a block diagram of a generic filtered SFBC-OFDM system with two transmit antennas and a single receiver antenna. Assume that M subcarriers in the range of M = {M0 , ..., M0 + M − 1} are assigned to a subband with a corresponding transmitter filter and receiver filter denoted as vector u and v in the time domain, respectively. A block of data symbol d = (d0 , d1 , ..., dM −1 )T is fed into the SFBC encoder with the k-th sub-block orthogonal code in the form of d2k d2k+1 , Dk = ∗ ∗ −d2k+1 +d2k where k = 0, 1, · · · , M/2 − 1, which generates two data sequences d(1) , d(2) as T T d(1) = d0 , −d∗1 , d2 , −d∗3 , ..., dM −2 , −d∗M −1 , d(2) = d1 , +d∗0 , d3 , +d∗2 , ..., dM −1 , +d∗M −2 . The two data streams are pre-equalized by M × 1 vector g before going through the IFFT/CP and filtering procedures independently as described in single-antenna systems, and they are then transmitted by the first and second antenna, respectively. A flat fading channel is considered for each subcarrier. Denote the channel impulse T (i) (i) (i) response between transmit antenna i as h(i) = h0 , h1 , · · · , h (i) , i ∈ {1, 2}, where Nch each and (i) hn is a complex Gaussian random variable with zero mean and variance (i) PNch (i) 2 n=0 E{|hn | } (i) = 1. ḧk = (i) PNch n=0 (i) 1 , (i) Nch +1 2π hn e−j N kn corresponds to the DFT of h(i) on the k-th subcarrier, which is a complex Gaussian random variables with zero mean n o (i) and variance one (E |ḧk |2 = 1). Assume that antennas at the transmitter are (1) (2) adequately apart so that channels are independent to each other, i.e., E{ḧk ḧk } = 49 4.3. Filter Selectivity Analysis and Discussion (1) (2) E{ḧk }E{ḧk }. Suppose that the length of CP/CS is chosen to ensure the nearly inference-free condition (4.16). After removing the CP/CS at the receiver, the FFT output, as the demodulated received signal vector, can be approximated according to (4.11) by r = VUH(1) Gd(1) + VUH(2) Gd(2) + Vw, (4.24) where the M × 1 vector r = (r0 , r1 , ..., rM −1 )T . The M × M matrices V = diag(v̈), U = diag(ü), H(i) = diag(ḧ(i) ), (i = 1, 2), u, v̈, ḧ(i) , are filter frequency response vectors of the transmitter filter, the receiver filter and the channels over the subaband of interest. w is a length N Gaussian random vector with w ∼ N(0M ×1 , N0 Im ). Extracting the pair of received data symbols indexed at 2k, 2k + 1 from (4.24) gives (1) (2) r2k = a2k d2k + a2k d2k+1 + v̈2k w2k , (1) (2) r2k+1 = −a2k+1 d∗2k+1 + a2k+1 d∗2k + v̈2k+1 w2k+1 , (i) (i) (4.25) i ∈ {1, 2}, j ∈ {2k, 2k + 1}. where aj = ḧj üj v̈j gj Assuming the channel and filtering information are perfectly known at the receiver, the following diversity combining scheme in the Alamouti SFBC decoder can be applied to give (1) (2) (a2k )∗ a2k+1 r2k = . (2) (1) ∗ dˆ2k+1 (a2k )∗ −a2k+1 r2k+1 dˆ2k (4.26) Substitute (4.25) into (4.26), we have (1) (2) dˆ2k = (|a2k |2 + |a2k+1 |2 )d2k + β2k d2k+1 + w2k , | {z } | {z } |{z} interference desired signal noise (2) (1) dˆ2k+1 = (|a2k |2 + |a2k+1 |2 )d2k+1 + β2k+1 d2k + w2k+1 , | {z } | {z } | {z } interference desired signal noise where (1) (2) (1) (2) β2k = (a2k )∗ a2k − (a2k+1 )∗ a2k+1 , (2) (1) (2) (1) β2k+1 = (a2k )∗ a2k − (a2k+1 )∗ a2k+1 , (1) (2) ∗ ∗ w2k = (a2k )∗ v̈2k v2k + a2k+1 v̈2k+1 w2k+1 , (2) (1) ∗ ∗ w2k+1 = (a2k )∗ v̈2k v2k − (a2k+1 )∗ v̈2k+1 w2k+1 . (4.27) 50 4.3. Filter Selectivity Analysis and Discussion The interference signals to d2k and d2k+1 can be easily found from (4.27), and the average interference power to d2k and d2k+1 can be computed as 2 ∗ σini,2k = E{β2k d2k+1 (β2k d2k+1 )∗ } = E{β2k β2k }σs2 , 2 ∗ σini,2k+1 = E{β2k+1 d2k (β2k+1 d2k )∗ } = E{β2k+1 β2k+1 }σs2 . (4.28) (i) We assume that the channel is constant over the two adjacent subcarriers, i.e., ḧk = (i) (i) (i) (i) (i) ḧk+1 (i = 1, 2). Applying ḧk = ḧk+1 and substituting aj = ḧj üj v̈j gj into (4.28), yields 2 2 σini,2k =σini,2k+1 (4.29) 2 = |ü2k |2 |v̈2k |2 |g2k |2 − |ü2k+1 |2 |v̈2k+1 |2 |g2k+1 |2 σs2 . Now, we divide the subcarrier set of interest M into two subsets: Mtrans and Mpass . Those subcarriers, locating in the passband, belong to the subset Mpass , and the other subcarriers in the transition bands on both sides of the filter are grouped into the subset Mtrans . We will study the spatial orthogonality with respect to subcarriers of two subsets separately. When both subcarriers 2k and 2k + 1 fall into the pass band region, the filter frequency response is constant. It gives |ü2k | = |ü2k+1 |, |v̈2k | = |v̈2k+1 | and |g2k | = |g2k+1 |, (M2k , M2k+1 ∈ Mpass ). (4.30) 2 2 substituting (4.30) into (4.29) results in zero interference power, σini,2k = σini,2k+1 = 0, and we have Proposition 4.3. The spatial orthogonality holds in SFBC-F-OFDM systems for those subcarriers in the passband region and the SNR is given as (1) γ2k = γ2k+1 = (2) |h2k |2 + |h2k |2 σs2 . σn2 2 (4.31) When at least one of the two subcarriers, either M2k or M2k+1 resides in the region of the transition band, where FFR selectivity occurs, and we have |ü2k | = 6 |ü2k+1 | and |v̈2k | = 6 |v̈2k+1 | (∀(M2k , M2k+1 ) ∈ Mtrans ). (4.32) 51 4.4. Numerical Results If no pre-equalization is implemented, g2k = g2k+1 = 1, we have Proposition 4.4. The spatial orthogonality is destroyed in SFBC-F-OFDM systems for those subcarriers in the transition band region due to the FFR selectivity, and the non-trivial interference power is quantified as 2 2 σini,2k = σini,2k+1 = |ü2k |2 |v̈2k |2 − |ü2k+1 |2 |v̈2k+1 |2 2 σs2 . (4.33) However, with the deployment of the pre-equalizer defined in (4.22) to reverse the effect of FFRS, the interference across all subcarriers can be forced to zero, and we have Proposition 4.5. The spatial orthogonality holds in SFBC-PF-OFDM systems with the implementation of the per-equalizer defined in (4.22), where the SNR is given as (1) (2) 2 |h2k |2 + |h2k | ρ2pre-equ σs2 γ2k = γ2k+1 = (1) . (4.34) 2 + |h(2) |2 v̈ 2 2σn2 |h2k |2 v̈2k 2k+1 2k Comparing (4.34) to (4.31), it can be seen that the SNR loss due to the per-equalizer in SFBC-PF-OFDM systems is Γ2k = 4.4 (1) (2) |h2k |2 + |h2k |2 (1) 2 |h2k |2 q̈2k + 2 ρ2pre-equ . (2) 2 2 |h2k | q̈2k+1 (4.35) Numerical Results In this section, we consider the evaluation of the following 1. the derived system model and Intra-NI single power induced by filters with different settings, 2. the bit error rate (BER) performance of F-OFDM single antenna systems under AWGN channels1 , and the different performance enhancement techniques represented in Section 4.2.5 and Section 4.3.1, 1 The reason that AWGN channels are chosen for verifying BER performance is to rule out the impact of the multi-path fading channel and focus on the interference produced by filtering. 4.4. Numerical Results 52 3. the BER performance of F-OFDM/PF-OFDM multi-antenna systems under multipath fading channels. The following parameters, unless otherwise specified, are adopted for simulations. The F-OFDM system occupies N = 1024 subcarriers. The considered multi-path fading channel of length Nf = 8 is a block-fading Gaussian channel (BFGC), where the duration of a transmitted data block is smaller than the coherent time of the channel. Therefore, the fading envelope is assumed to be constant during the transmission of a block and independent from block to block. The length of a block (a frame) lasts over a duration of 14 OFDM symbols. It is assumed that the channel state information (CSI) is perfectly known at the receiver. A soft-truncated sinc filter defined in [75] is employed at the transmitter and receiver side, with filter length of Nu = Nv = N/2 and slope controlling parameters αu = 0.6, αv = 0.65. 4.4.1 Numerical Results for Intra-NI and FFRS The derivation of Intra-NI in subsections 4.2.2 - 4.2.4 is numerically evaluated and plotted for different values in BWP width and CR length. Fig. 4.5 shows an example of power distributions for the desired signal and interference signal (ICI,f-ISI, and b-ISI) on a subcarrier level, evaluated through (4.14) with FFT size of 1024, BWP width of 36 subcarriers in (a) and 240 subcarriers in (b). In addition, no CP/CS was added for interference alleviation. It is shown in the figure that the theoretical value of Intra-NI power matches the simulation result. It is clearly visible that the interferences in the wider subband have lower power as a whole than that of in the narrower band. Fig. 4.5 (a) indicates that uneven power is distributed for both the desired signal and interference signal among subcarriers. Comparing to other subcarriers, those near the edge (edge subcarriers) have lower desired signal power while experiencing higher interference. The two overlapping curves corresponding to the interference power from the previous F-OFDM (f-ISI) and next F-OFDM symbol (b-ISI) indicates the same power distribution due to the symmetry of the filters. The trends are also captured in Fig. 4.5 (b) for the wider subband. 53 4.4. Numerical Results (a) (b) 0 desired sig. (sim.) desired sig. (anal.) total int. (sim.) total int. (anal.) ICI (anal.) forward ISI (anal.) backward ISI (anal.) -5 -10 Power (dB) -15 -20 -25 -30 -35 -40 -45 10 20 SC index 30 50 100 150 SC index 200 Figure 4.5: Power of desired signal and Intra-NI signal components with N = 1024, Ncr = 0, Nu = Nv = 512, αu = 0.6, αv = 0.65. The three contributions (ICI, f-ISI, and b-ISI) to the total interference are evaluated individually from our analytical expressions, which can not be fulfilled through simulation, thus only the analytical results of them are plotted. (a) A narrow subband of 3 RBs (36 subcarriers). (b) A wider subband of 20 RBs (240 subcarriers). 54 4.4. Numerical Results ICI ISI Power (dB) max avg. min BWP width (RBs) Figure 4.6: Max, min, and average normalized power of ICI/ISI with respect to subcarriers against subbands of different width with N = 1024, Ncr = 0, Nu = Nv = 512, αu = 0.6, αq v = 0.65, ISI represents either forward ISI or backward ISI. All curves of the figure are generated from the analytical results. 55 4.4. Numerical Results Avg. intraNI power (dB) 1RB 2RBs 5RBs 10RBs 20RBs 50RBs CR length (samples) Figure 4.7: Average effective interference power with respect to subcarriers vs. number of CRs with N = 1024, Ncr = 0, Nu = Nv = 512, αu = 0.6, αv = 0.65, the solid black dots indicate the number of CRs equals the width of the filter main lobe of the corresponding subband. All curves of the figure are generated from the analytical results. 4.4. Numerical Results 56 The maximum, minimum, and average normalized power of ICI/ISI with respect to subcarriers of the same subband are shown in Fig. 4.6, where the ISI represents forward ISI or backward ISI as both have the same power distribution indicated in Fig. 4.5. These results provide a direct comparison among BWPs with varying width from 1 RB to 50 RBs. It can be seen that the maximum normalized power keeps constant as the width of subband grows. However, this is not the case with the minimum and average normalized power, where both decrease as the BWP width increase. This implies that narrower BWPs are more prone to Intra-NI compared to wider BWPs. These features apply to both the ICI and ISIs. Fig. 4.7 presents the average effective interference power of six subbands of variable width versus the number of CRs, where Ncp = Ncs = Ncr /2. The effective interference here refers to the sum of the ICI, f-ISI, and b-ISI. The effect of CR length for alleviating Intra-NI interference is observed from all these curves. The power of the average effective interference decreases as the length of CR increases, and it drops under 25dB for all six subbands when the number of CRs equals to the length of corresponding filter main lobe due to the fact that most of the filter energy is contained in the main lobe. The BER performance of F-OFDM under the AWGN channel is evaluated and plotted in Fig. 4.8. The results are presented in two cases, Ncr = 0 in (a) and Ncr = 72 in (b), each having six curves corresponding to a different BWP width and a curve representing the BER of legacy OFDM for a benchmark comparison. Taking into the consideration of computational complexity at the receiver side, one-tap equalization method is adopted. When the Intra-NI is not handled by introducing CR, the performance of F-OFDM, as shown in Fig. 4.8(a), degrades dramatically comparing with OFDM systems. Moreover, error floors also tend to develop for all subbands. Another interesting observation is that the performance degradation in narrower subbands is higher than that of in the wider subbands, again suggesting that narrower subbands suffer more Intra-NI distortion. In (b), the BER performance is significantly improved due to the use of the CR. There is still a gap of approximately 2-5 dB, subject to how wide the subband of interest is, which implies that there is still space to improve, especially for narrow subbands. The effect of different approaches to interference suppression is shown in terms of BER 57 4.4. Numerical Results (a) (b) f-OFDM (1RB) f-OFDM (2RBs) f-OFDM (5RBs) f-OFDM (10RBs) f-OFDM (20RBs) f-OFDM (50RBs) OFDM 10-1 BER 10-2 10-3 10-4 10-5 0 5 10 14 0 EbNo (dB) 5 10 14 EbNo (dB) Figure 4.8: Error performance for F-OFDM systems under AWGN channel with QPSK modulation. (a) Ncp = Ncs = 0. (b)Ncp = Ncs = 36. 58 BER 4.4. Numerical Results one-tap equ. (0CR) ZF(0CR) MMSE(0CR) BwPIC (0CR,1Iter) BwPIC (0CR,2Iters) one-tap equ. (24CR) ZF(24CR) MMSE(24CR) BwPIC (24CRs,1Iter) BwPIC (24CRs,2Iters) EbNo (dB) Figure 4.9: Error performance comparison with and without implementation of BwPIC for FOFDM systems under AWGN channels with QPSK modulation. 59 4.4. Numerical Results 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 1 1 0.5 0.5 0 0 0 Figure 4.10: Interference power (normalized by signal power) versus filter frequency responses of consecutive subcarriers. performance enhancement in Fig. 4.9 with two different CR length setting, Ncr = 0 and Ncr = 24 in a subband of 12 subcarriers. It can be seen that ZF and MMSE have almost the same BER performance in both CR settings. However, the BER performance with the implementation of BwPIC improves significantly compared to the systems without BwPIC, and the results also show that the algorithm converges with no more than two iterations. It is worth mentioning that the performance gain comes at the cost of increasing computational complexity due to the related operations involved in Algorithm 1. 4.4.2 Numerical Analysis for Filter Selectivity Fig. 4.10 plots the interference power for an SFBC-FOFDM system without the preequalization obtained in (4.33). The result shows that the interference increases as the difference in filter gain between two subcarriers grows, implying that the system suffers more interference in a region with higher filter selectivity. The BER performance of F-OFDM in single and multiple antenna systems is numerically evaluated under the AWGN and multi-path fading channel respectively. In the case of single antenna systems, we use the same simulation parameters chosen in 4.4.1 so that the performance with and without pre-equalization can be compared fairly. Fig. 4.11 compares the BER performance of F-OFDM systems with and without pre- 60 4.4. Numerical Results f-OFDM (1RB) f-OFDM (5RBs) f-OFDM (20RBs) pf-OFDM (1RBs) pf-OFDM (5RBs) pf-OFDM (20RBs) OFDM -1 10 Bit Error Rate 10-2 10-3 10-4 10-5 0 5 10 EbNo (dB) Figure 4.11: Error performance comparison with and without implementation of preequalization under AWGN channels with QPSK modulation. equalization, and it can be clearly seen that PF-OFDM outperforms F-OFDM for all subbands. It is also observed that the BER performance of PF-OFDM is very close to OFDM when the width of the subband is over 5 RBs. Although there is still some gap for the narrower subbands, the performance is considerably improved in comparison to that of PF-OFDM without pre-equalization. The remaining performance gap to OFDM systems can be explained by the power loss due to the pre-equalization. The BER performance of SFBC-OFDM is numerally evaluated and plotted in Fig. 4.12 for different values of BWP width with and without pre-equalization. It can be seen from the figure that SFBC-PF-OFDM dramatically outperform SFBC-F-OFDM, and as the width of subband grows, the BER performance converges to the benchmark result of SFBC-OFDM. The effect of different modulation scheme is also observed from these curves, and the BER performance of QPSK is better than 16-QAM as expected. In the 61 4.4. Numerical Results (a) (b) 10-1 BER 10-2 10-3 pf-OFDM (1 RB) f-OFDM (1 RB) pf-OFDM (2 RBs) f-OFDM (2 RBs) pf-OFDM (3 RBs) f-OFDM (3 RBs) pf-OFDM (5 RBs) f-OFDM (5 RBs) pf-OFDM (10 RBs) pf-OFDM (10 RBs) OFDM 10-4 10-5 0 5 10 15 20 EbN0 (dB) 24 0 5 10 15 20 24 EbN0 (dB) Figure 4.12: BER performance for filtered SFBC-OFDM systems with and without pre-equalization under Rayleigh fading channel with N = 1024, Nu = Nv = 512, αu = 0.6, αv = 0.65. (a) QPSK. (b) 16-QAM. 4.5. Summary 62 case of 16-QAM, error floors are quickly developed due to the interference introduced by spatial non-orthogonality in SFBC-F-OFDM without pre-equalization systems, implying that it cannot be implemented when higher modulation schemes are adopted. However, error floors do not exist in SFBC-PF-OFDM with pre-equalization systems as the spatial orthogonality is protected from being destroyed by the FFR selectivity. 4.5 Summary Based on the framework developed in Chapter 3, the analytical expressions of the Intra-NI signal, including ICI, forward ISI, and backward ISI, were derived. The IntraNI-free condition for F-OFDM systems was developed. In addition, we proposed a low-complexity FEQ algorithm - BwPIC to cancel the Intra-NI. Furthermore, the effect of FFRS to single antenna and multi-antenna F-OFDM systems was discussed, and a pre-equalization approach was proposed to tackle it. As the simulation results show, the analytical interference power calculated from analytical expression matches the simulation results, validating the analytical model established in this thesis. The proposed BwPIC effectively cancels the interference signal and significantly improves the BER performance. With the proposed equalizer, PF-OFDM outperforms F-OFDM and is close to OFDM in single antenna systems. In contrast, in multi-antenna systems, it protects spatial orthogonality from destruction by the FFRS. The work presented in this chapter provides a useful reference and valuable guidance for the practical deployment of this waveform in 5G wireless systems. Chapter 5 Inter-Numerology Interference Analysis For some use-cases, mixing of different numerologies on the same carrier frequency may be beneficial, e.g., to support different services with very different latency requirements. In an OFDM-based system with different numerologies (subcarrier bandwidth and/or cyclic prefix length) multiplexed in frequency-domain, only subcarriers within a numerology are orthogonal to each other. Subcarriers of one numerology interfere with subcarriers of another numerology, since energy leaks outside the subcarrier bandwidth and is picked up by subcarriers of the other numerology. In this Chapter, we first derive the analytical expression of the Inter-NI and its power in multi-numerology systems, and the usages that the analytical work enables are then discussed. Finally, a study case utilizing the offered analysis on power allocation is provided, where a optimization problem of maximizing sum rate is formulated, and a solution is also given. 63 64 5.1. Inter-Numerology Interference Analysis 5.1 Inter-Numerology Interference Analysis The Inter-NI to a numerology refers to the signal components contributed by other numerologies. When a multiplexing signal is generated from several different numerologies, the orthogonality between subcarriers no longer holds. Subcarriers from one numerology interfere with those from the others. Subcarriers of one numerology may not fit in zeros of sinc response of subcarriers of another numerology as illustrated in Fig. 5.1, where a numerology based on subcarrier spacing ∆f2 interferes with another numerology based on subcarrier spacing ∆f1 . It worth mentioning that even subcarriers of Numerology 2 lie in the zeros of sinc response of subcarriers of Numerology 1, it doesn’t necessarily mean that Numerology 1 doesn’t interfere with numerology 2 due to the fact that the two numerologies have different symbol length. Only half of a symbol of Numerology 1 fits into a FFT window of Numerology 2 when FFT is performed at a receiver of Numerology 2, which inevitably causes spectrum shift, i.e., the zeros of sinc response of subcarriers of Numerology 1, deviate from its original position which inflicts the interference to Numerology 2. When mixed numerologies are implemented in CP-OFDM systems, the energy leaked from one numerology to the other, a.k.a, OOB emission, is high, due to the spreading characteristic of sinc filters. While in F-OFDM systems, each numerology is accommodated in a subband with a better-localized filter so that the level of interference between different numerologies can be reduced. We will give the analytical expression of the Inter-NI signal in multi-numerology systems in the sequel. 5.1.1 The Expression of the Inter-NI Signal The contribution from the j-th numerology to the received signal of the i-th numerology (i←j) in the k-th OFDM window at the receiver, denoted as zk (i←j) zk , can be approximated as (i−) V(i:m) H(i:m) C(i←j) U(j:m) x(i←j) , j ∈ Snum k k ≈ , V(i:m) H(i:m) blkdiag U(j:m) , ν (j) x(i←j) , j ∈ S(i+) num k ν (i) (5.1) 65 5.1. Inter-Numerology Interference Analysis Numerology 1 ∆f1 Amplitude Numerology 2 ∆f2 3 × ∆f1 Frequency Inter-numerology interference Figure 5.1: Inter-numerology interference. where (i←j) xk x(j)ν (j) , dk (i) e # = " ν T T T T (j) (j) (j) , , · · · , x (k+1)ν (j) x kν (j) , x kν (j) +1 −1 ν (i) ν (i) ν (i) (i−) j ∈ Snum j∈ . (5.2) (i+) Snum The detailed derivation can be found in appendix C. After the CP removal and the FFT, the interference symbol vector from the j-th numerology to the i-th numerology in the frequency domain can be written as (i←j) (i←j) yk,inter = (F(i) )H R(i) cr zk . (5.3) After conducting some substitutions based on (3.4), (5.1), and (5.2), we can express (i←j) the Inter-NI symbol vector yk,inter in (5.3) as (i←j) (i←j) (i←j) dk . yk,inter = Ψk (5.4) where (i←j) Ψk = (i) H (i) (i:m) H(i:m) C(i←j) U(j:m) T(j) F(j) , ρ(j) cr (F ) Rcr V cr k j ∈ Snum ρ(j) (i) H (i) (i:m) H(i:m) blkdiag U(j:m) T(j) F(j) , ν (j) , cr (F ) Rcr V cr ν (i) j ∈ Snum (i−) (i+) (5.5) 66 5.1. Inter-Numerology Interference Analysis and (i←j) dk (j) (j) d ∈ CM ×1 , dk ν (j) e (i) # = " ν T T T T (i) (j) (j) (j) ∈ CM ×1 , d kν (j) , d kν (j) +1 , · · · , d (k+1)ν (j) −1 ν (i) ν (i) ν (i) (i−) j ∈ Snum j∈ . (i+) Snum (5.6) (i←j) The matrix Ψk (j) ∈ C M (i) ×(d ν (i) eM (j) ) ν can be interpreted as the equivalent channel which transforms data symbols from the j-th numerology into the InterNI to the i-th (i←j) numerology in the k-th OFDM symbol. Specifically, the (r, c)-th element of Ψk (i←j) (i←j) Ψk,r,c , is the complex channel gain of r-th symbol of dk (i←j) BWP(i) . The vector dk ∈C (j) (d ν (i) eM (j) )×1 ν , on the c-th subcarrier of comprises those data symbols from j-th numerology which generate interference to the symbols in the k-th OFDM window of the i-th numerology. Adding interference components from all the other numerologies yields the expression of total Inter-NI to the i-th numerology as (i) yk,inter = X (i←j) (i←j) dk . Ψk (5.7) j∈Snum \{i} 5.1.2 The Analytical Expression of the Inter-NI Power The instantaneous power of the Inter-NI signal contributed from the j-th numerology to all the subcarriers in the k-th OFDM symbol of the i-th numerology can be grouped into an M (i) dimensional vector as (i←j) (i←j) (i←j) (i←j) H (i←j) H |yk,InterNI |2 = diag Ψk dk (dk ) (Ψk ) . (i←j) The above expression for the Inter-NI and the expression of Ψk (5.8) in (5.5) indicate that some parameters could affect the level of the interference. Those parameters include BWP bandwidth/guard band M (i) , M (j) , G(i) , G(j) included in F(i) ,F(j) , pulse shaping filter parameters v(i) ,u(j) , channel gain h(i) , and subcarrier power of inter(j) fering numerologies (σs )2 . The average level of distortion caused by the interfering 67 5.1. Inter-Numerology Interference Analysis numerologies can be expressed as a function of these parameters as (i←j) |ȳk,InterNI M (a) , G(a) , u(j) , v(i) , h(i) , (σs(j) )2 |2 n o (i←j) (i←j) H = E diag Ψk (Ψk ) (σs(j) )2 , a ∈ {i, j}, (5.9) The identical power spectrum distribution (PWD) property of fading channel, i.e. E ḧḧH = I, indicates that the interference power in (5.9) can be rewritten as (i←j) |ȳk,InterNI M (a) , G(a) , u(j) , v(i) , h(i) , (σs(j) )2 |2 (i←j) (i←j) H (j) 2 = diag Ψ̃k (Ψ̃k ) (σs ) , a ∈ {i, j}, (5.10) where (i←j) Ψ̃k = (i) H (i) (i:m) C(i←j) U(j:m) T(j) F(j) , ρ(j) cr (F ) Rcr V cr k ρ(j) (i) H (i) (i:m) blkdiag U(j:m) T(j) F(j) , ν (j) , cr (F ) Rcr V cr ν (i) (i−) j ∈ Snum j∈ . (5.11) (i+) Snum The signal to Inter-NI ratio (SIR), as a metric to quantify the distortion level, can then be computed according to (4.11) and (5.10) as (i←j) SIRk,InterNI M (a) , G(a) , u(a) , v(i) , (σsa )2 diag (Λ(i) )H Λ(i) , a ∈ {i, j}. = (i←j) (i←j) H (i←j) diag Ψ̃k (Ψ̃k ) ξ (j) (5.12) (i) where ξ (i←j) = (σs )2 /(σs )2 defines the power offset factor between the j-th numerology and the i-th numerology. It is worth pointing out that the Inter-NI level for multi-numerology OFDM systems can be obtained by replacing filtering matrices with identity matrices. Therefore, our analytical work is generic for both OFDM and F-OFDM. 5.1.3 Further Discussion on Inter-NI The level of distortion induced by mixed numerologies, analytically expressed as a function of several system parameters in (5.10), can be exploited to analyze and evaluate these parameters, i.e., the minimum guard band, the minimum CR length, or the minimum filter length, etc., to meet given targets with respect to the maximum level of 5.2. A Case Study: Power Allocation in the Presence of Mixed Numerologies 68 distortion. In addition, the expressions of instantaneous power of the Inter-NI in (5.8), facilitates the formulation of optimization problems of spectrum/power efficiency for multi-numerology systems. It is impossible to investigate all these usages that our analysis work enables in one paper due to the page limitation. However, we will investigate how the offered analytical work is utilized to formulate an optimization problem on power allocation in single-user multi-numerology F-OFDM/OFDM systems. Our work will be extended to optimizing power and subcarrier allocation in multi-user systems in the future. 5.2 A Case Study: Power Allocation in the Presence of Mixed Numerologies In this section, we provide a case study on optimizing power allocation for single-user multi-numerology systems. Assume that subcarrier and guard allocation for the numerologies have been solved. We consider a block fading channel which is assumed to be constant during the transmission of a block and independent from block to block. The length of a block lasts over a duration of several OFDM symbols of the greatest length among all the numerologies. Power is allocated on a block basis in the BS. We define a T M × M (i) dimensional power allocation vector p = (p(0) )T , (p(1) )T , · · · , (p(M −1) )T , (i) (i) (i) where the M (i) dimensional vector p(i) = [p0 , p1 , · · · , pM (i) −1 ]T corresponds to the power allocation in BWP(i) . According to (3.35), we can express the QAM symbol received on the m-th subcarrier in the k-th OFDM symbol of BWP(i) as (i) (i) (i) (i) <i> yk,m = yk,m,des + yk,m,intra + yk,m,inter + ŵk,m . (i) (i) (5.13) (i) According to section 4.1, ŵk,m is an AWGN with ŵk,m ∼ N(0, ηm,noise ). Assume that the length of CR is sufficiently long to satisfy the near IntraNI-free condition discussed in (4.16). According to (4.11) and (5.7), we have the desired signal and the interference 5.2. A Case Study: Power Allocation in the Presence of Mixed Numerologies 69 signal as (i) yk,m,intra ≈ 0, (i) yk,m,des = Λ(i) m,m q (i) (i) pm d¯k,m , (j) d ν (i) eM (j) −1 ν (i) X X j∈Snum \{i} n=0 yk,m,inter = (i←j) Ψk,m,n q (j) (i←j) pn mod M (j) d¯k,n , (5.14) (i) (i←j) (i) (i←j) where d¯k,m and d¯k,m are normalized data symbols of dk,m and dk,m , respectively. The SINR on the m-th subcarrier in BWP(i) can be expressed as a function of p, which is written as (i) (i) γm (p) = (i) |Λm,m |2 pm (j) d ν (i) eM (j) −1 (5.15) ν X X j∈Snum \{i} n=0 (i←j) (j) (i) |Ψk,m,n |2 pn mod M (j) + ηm,noise ∆f (i) We consider continuous bit-loading and write the achievable bit rate on the m-th subcarrier in BWP(i) as (i) (i) (i) rm (p) = ∆f log 1 + γm (p) (5.16) in the unit of bps (bit per second). The achievable rate for BWP(i) can then be computed as X R(i) (p) = X (i) rm (p) = m∈M(i) (i) (i) ∆f log 1 + γm (p) (5.17) m∈M(i) bps per channel use. Problem formulation: Our optimization problem seeks to maximize system sum rate subject to a maximum power constraint. The problem is written as max p≥0 s.t. X i∈Snum X X ω (i) (i) (i) ∆f log(1 + γm (p)) (5.18) m∈M(i) X i∈Snum m∈M(i) (i) pm ≤ P0 , Where P0 is the maximum transmission power of the system, and each ω (i) is a tunable non-negative weight that allows a trade-off between the rates allocated to the numerologies. Equivalently, these weights allow the system operators to assign a different QoS level to each numerology. 70 5.2. A Case Study: Power Allocation in the Presence of Mixed Numerologies (i) Replacing γm (p) in (5.18) with its expression in (5.15) reveals the objective function as a difference of concave (d.c.) function in p. The problems with d.c. structure can be shown [83] as NP-hard, and a global optimal solution is difficult to achieve. The Iterative water falling (IWF) approach [84] achieves an approximate solution by considering this problem as M isolated sub-problems and iterate them until convergence, in which each sub-problem optimizes power allocation p(i) by treating all other power p(j6=i) as fixed noise. We will introduce the approach described in [85] to our power allocation scheme to deal with the d.c. structure and relax the non-convex problem (5.18). The following lower bound is leveraged for relaxing the above non-convex problem α log z + β ≤ log(1 + γ). (5.19) It is tight at a chosen γ when the constant {α, β} specified as α = γ 1+γ . β = log(1 + γ) − γ log γ 1+γ (5.20) Applying (5.19) to the optimization problem specified in (5.18) results in a relaxed problem max p≥0 s.t. X X (i) ω (i) ∆f (i) (i) (i) αm log(γm (p) + βm (5.21) i∈Snum m∈M(i) X X i∈Snum m∈M(i) (i) (i) pm ≤ P0 , (i) where the vaule of αm and βm are fixed for a given p. However, the relaxed problem (i) remains not convex with respect to p because SINR function γm (p) is not convex. A further variable substitution p = ep̃ convert the optimization problem into a new function of a variable p̃ as max p̃≥0 s.t. X X (i) (i) (i) p̃ (i) ω (i) ∆f (αm log(γm (e ) + βm ) i∈Snum m∈M(i) X X i∈Snum m∈M(i) (i) ep̃m ≤ P0 , (5.22) 71 5.3. Numerical Results (i) Expanding the term of log γm (ep̃ ) yields the following expression (i) p̃ (i) log γm (e ) = 2 log |Λ(i) m,m | + p̃m (j) d ν (i) eM (j) −1 − log ν X X j∈Snum /{i} n=0 (i←j) {p̃ |Ψk,m,n |2 e (j) n mod M (j) } (i) + (ηm,noise )∆f (i) , (5.23) which comprises a sum of a linear term and a convex log-sum-exp term. This proves the convexity of the objective function in (5.22). The constraint function being a (i) sum of convex terms (eũm ), thus is also convex. The above analysis concludes that (5.22) is a convex optimization problem, thus can be solved through a standard convex optimization package like CVX [86]. Here, we maximize a lower bound on the achieving sum rate. The bound can then be improved iteratively, which yields the following algorithm. Algorithm 2 power allocation 1: Inputs: pinit , Snum , M(i) , Λ(i) , Ψ(i) , i ∈ ∀Snum 2: output: optimal power allocation p∗ 3: Initialize iteration counter t = 0 4: p<t> = pinit , p̃<t> = log(p<t> ) 5: repeat 6: 7: (i) update (γm )<t> using (5.15) for all i ∈ Snum and all m ∈ M(i) (i) (i) update (αm )<t> , (βm )<t> using (5.20) for all i ∈ Snum and all m ∈ M(i) 8: solve convex problem (5.22) to obtain solution p̃<t> 9: p<t> = e(p̃ 10: <t> ) t←t+1 11: until Convergence 12: return p(∗) = p<t> 5.3 Numerical Results In this section, the impact from several system parameters on the level of distortion will be examined through the analytical studies conducted in Section 5.1. Moreover, 72 5.3. Numerical Results parameters Table 5.1: Numerology related parameters BWP(a1 ) BWP(a0 ) BWP (a2 ) µ 2 1 0 subcarrier spacing (kHz) 60 30 15 FFT size 256 512 1024 0-360 kHz (5 PRBs) 360-540kHz (5 PRBs) 540 - 630 kHz (5 PRBs) BWP allocation the performance of the proposed power allocation scheme in Section 5.2 is also evaluated via Monte Carlo simulations. Three BWPs (BWP(a0 ) , BWP(a1 ) , BWP(a2 ) ) with different numerologies are considered in a system with total bandwidth of 1024 × 15 kHz. All BWPs are assumed to have 5 physical resource blocks (PRBs) equivalent to 60 subcarriers allocation, and guard bands are allocated within BWPs. The level of distortion and BER are always evaluated for BWP(a0 ) while signals from BWP(a1 ) and BWP(a2 ) serve as Inter-NI sources. Table 5.1 lists numerology-related parameters. In the setting, signal from BWP(a1 ) with a wider subcarrier spacing interferes with BWP(a0 ) from the left-hand side, while signal from BWP(a2 ) with a narrower subcarrier spacing interferes from the right-hand side. The AWGN noise power density is assumed to be -174 dBm/Hz. No other interference sources are assumed and the IntraNI is well eliminated through added cyclic redundancy. Thus, the term “interference” below refers to Inter-NI between different BWPs. In the case of F-OFDM, soft-truncated sinc filters defined in [75] are implemented at the transmitter and receiver. The roll-off factor of all used time domain window is fixed at 0.6, while different filter lengths are employed to investigate the impact from filters. The length of a filter is measured as the number of sidelobs (SLs) it has on each side. For simplicity, we assume that matched filters are used in all cases except the one which uses unmatched filter to specifically investigate the impact from transparent filtering in Fig. 5.4. The SIR expression developed in (5.12) is evaluated with different settings to study the impact that guard bands and filters have on the Inter-NI in Fig. 5.2-5.4. Some Monte Carlo simulations are conducted to validate our analytical work, and the matching result suggests that our derivations are valid. It can be observed from all the tree figures that 73 5.3. Numerical Results (a) 60 (b) 50 SIR(dB) 40 30 20 0 gb (anal.) 0 gb (sim.) 60 kHz gb (anal.) 60 kHz gb (sim.) 120 kHz gb (anal.) 120 kHz gb (sim.) 10 0 0 4 8 12 16 20 23 36 40 44 SC index Figure 5.2: 48 52 56 59 SC index SIR with different guard band settings on interfering BWPs. All filters’ length = 3 SLs, no guard band in BWP(a0 ) , Axises truncated to show 24 subcarriers to the interference source side, gb refers to guard band. (a) Interference source: signal of BWP(a1 ) . (b) Interference source: signal of BWP(a2 ) . 74 5.3. Numerical Results (a) 50 (b) 45 40 35 SIR (dB) 30 25 20 15 OFDM (anal.) OFDM (sim.) f-OFDM: 1SL (anal.) f-OFDM: 1 SL (sim.) f-OFDM: 3 SLs (anal.) f-OFDM: 3 SLs (sim.) f-OFDM: 5 SLs (anal.) f-OFDM: 5 SLs (sim.) 10 5 0 0 4 8 12 SC index Figure 5.3: 16 20 23 36 40 44 48 52 56 59 SC index SIR with different settings on the length of filters of interfering sources (matched filtering scenario). No guard band is used. The length of filters in BWP(a0 ) is fixed at 3 SLs, while the filters in BWP(a1 ) and BWP(a2 ) takes three different length: 1 SLs, 3SLs and 5 SLs. Axises truncated to show 24 subcarriers to the interference source side. (a) Interference source: signal of BWP(a1 ) . (b) Interference source: signal of BWP(a2 ) . 75 5.3. Numerical Results (a) 50 (b) 45 40 35 SIR (dB) 30 25 20 15 OFDM (anal.) OFDM (sim.) f-OFDM: 1SL (anal.) f-OFDM: 1 SL (sim.) f-OFDM: 3 SLs (anal.) f-OFDM: 3 SLs (sim.) f-OFDM: 5 SLs (anal.) f-OFDM: 5 SLs (sim.) 10 5 0 0 4 8 12 SC index 16 20 23 36 40 44 48 52 56 59 SC index Figure 5.4: SIR with different settings on the length of filters of interfering sources (transparent filtering scenario). No guard band is used. The length of filters at the transmitter is fixed at 3 SLs, while the filters at the receiver takes three different length: 1 SLs, 3SLs and 5 SLs. Axises truncated to show 24 subcarriers to the interference source side. (a) Interference source: signal of BWP(a1 ) . (b) Interference source: signal of BWP(a2 ) . 5.3. Numerical Results 76 the level of interference decreases (SIR increases) as subcarriers move away from the interference sources. In OFDM, the SIR curves climbs very slowly over the subcarrier, and it finally stays a level at about 20dB and 30dB for BWP(a1 ) , and BWP(a2 ) as the interference source, respectively, which suggests that subcarriers of a numerology suffer more interference from a source of wider subcarrier spacing. In contrast, the SIR curves of F-OFDM rise much more rapidly over the subcarrier, and the level of interference is more independent to numerology as curves corresponding to different interference source have similar growing rate given a same setting. When we compare the SIR curves between OFDM and F-OFDM, we can easily see that the latter performs much better except the first several subcarriers which only show relatively small improvement. This implies that the level of interference can be controlled below a pre-defined value with the employment of filtering as well as guard bands which ditch the first few subcarriers. The effect of guard band on the level of interference in terms of SIR is illustrated in Fig. 5.2. The two filters at the receiver and transmitter in F-OFDM are assumed to have 3 SLs. No guard band is adopted for BWP(a0 ) , while three different guard band sizes (0, 60 kHz, 120 kHz) are considered for interfering BWPs. As a guard band becomes wider, we observe that the level of the interference reduces (SIR increases) for OFDM and F-OFDM. However, the effect is much more significant in F-OFDM than its peer in which the SIR improvement becomes marginal as the subcarriers move away from the interference sources. Moreover, when it comes to different numerologies, we found that guard band functions more effectively in terms of reducing interference for the BWP with a smaller subcarrier spacing in OFDM by comparing 5.2(a) with Fig. 5.2(b). However, this trend is not well reproduced in F-OFDM due to the additional signal processing. In Figs. 5.3 and 5.4, we study how filters affect the level of interference in the case of matched filtering and transparent filtering, respectively. Fig. 5.3 shows the SIR change of BWP(a0 ) , when filters of different length in interfering BWPs are considered. Matched filters are assumed for all BWPs, and the length of the filters used in BWP(a0 ) is fixed. It is clearly visible that the interference decreases as the length of filters in interfering sources increases. This can be well explained by the fact that longer filters enjoy better frequency localization. Fig. 5.4 describes the impact on the interference 77 5.3. Numerical Results (a) (b) 10-1 -2 Bit Error Rate 10 -3 10 10-4 0 5 10 15 0 EbN0 (dB) 5 10 15 EbN0 (dB) Figure 5.5: BER performance with different settings on power offset under AWGN channel and 16QAM modulation scheme. One guard subcarrier is implemented on (a ) each side of all three BWPs. Power offset factor ζ is defined as ζ = ( σσ(v)i )2 dB, i = 1, 2. (a) Interference source: signal of BWP(a1 ) . (b) Interference source: signal of BWP(a2 ) . from transparent filtering (unmatched filters between receivers and transmitters of all three BWPs). The length of the filters at the transmitters are configured at 3SLs, while the receives have three different settings on filter length: 1SLs, 3SLs, and 5SLs. When the filter length grows, we see the similar trend occurring in the matched filtering case, which suggests the feasibility of transparent filtering. Fig. 5.5 shows BER performance of BWP(a0 ) under different settings on power offset with AWGN channel and 16QAM modulation. We notice that the dashed curves, corresponding to different power offset in OFDM, are well apart from each other, and error floors are quickly developed especially for higher power offset cases. The significant BER degradation each 3dB increase in power offset suggests that power offset has a 78 5.3. Numerical Results (a) 4.5 9 4 8.5 3.5 8 spectrual efficency (bits/s/Hz) (b) 9.5 3 OFDM (zero INI) f-OFDM (proposed) f-OFDM (IWF) f-OFDM (equ.) OFDM (proposed) OFDM (IWF) OFDM (equ.) 7.5 2.5 7 6.5 2 6 1.5 5.5 1 5 0.5 0 3 6 9 SNR (dB) 12 15 4.5 18 21 24 SNR (dB) 27 30 Figure 5.6: Spectrum efficiency comparison among different power allocation schemes under fading channel in the presence of mixed numerologies: proposed, IWF, and equal power allocation. (a) Lower SNR region. (b) Higher SNR region. 5.4. Summary 79 great impact on the interference between different numerologies in OFDM. In contrast, the BER degradation due to power offset are much lower in F-OFDM. This implies that F-OFDM systems are more resilient to power offset than OFDM systems, which again conforms the importance of spectrum confinement techniques to the multi-numerology systems. When we compare interference from different numerologies between Fig. 5.5 (a) and (b), we find that BWP(a0 ) suffers more from the interference source of wider subcarrier spacing in the presence of power offset. The spectrum efficiency (SE) of the proposed power allocation in section 5.2 was numerically compared with other two schemes, IWF and equal power allocation, in FOFDM/OFDM multi-numerology systems in Fig. 5.6. The extended typical urban (ETU) channel defined in [87] is considered. Without loss of generality, we assume all BWPs are equally weighed and no guard band is used. We observe that the SE of all schemes are very close in Fig. 5.6 (a), while the SE of the proposed power allocation method stands out from the others in Fig. 5.6 (b). This concludes that power allocation works more effectively in higher SNR region where the transmission tends to be interference-limited. It worth to mention that F-OFDM performs generally better than OFDM in all schemes. However, the SE improvement with the proposed scheme is higher in OFDM. This can be explained that there is more Inter-NI in OFDM; Hence, there is more room to improve through power allocation. 5.4 Summary The inter-numerology interference was analyzed based on the model developed in Chapter 3. The analytical metric for quantifying the strength of the distortion was derived as a function of several system parameters. The impact of these parameters on the inter-numerology interference were investigated analytically and numerically. The usages of analytical expression inter-numerology interference are discussed. A case study on optimizing power allocation based on the offered analysis was also presented. This work conducted in this chapter provides an analytical guidance on the system design in support of 5G multi-service transmission over a unified physical infrastructure. Chapter 6 Conclusions and Future Works The wireless communication environments of 5G and beyond networks are highly heterogeneous. To efficiently support this heterogeneity, 5G NR should be configurable to a very high extend. On the physical layer, this configurability includes mixed numerology multiplexing to accommodate diversified services with different technical requirements. Multiplexing different numerologies over same baseband inevitably inflicts mutual interference between them, which calls for waveforms with better spectral localization to better isolate a service from others. This thesis has been focused on addressing the coexistence/isolation issues of multiple services. In what follows, the summary and conclusions of the research will be first presented. Then, taking into consideration of the state-of -the-art on mixed numerology multiplexing presented in Chapter 2, some future research directions will be discussed. 6.1 Summary and Conclusions In this thesis, we first presented a comprehensive literature review in Chapter 2 on the efforts to address the challenges from the coexistence/isolation of multiple services over a unified physical layer from the perspective of waveform and numerology. To the best of our knowledge, a comprehensive analysis of all factors contributing to the interferences, within a numerology and between numerologies, in mutli-numerology FOFDM systems is still lacking. 80 6.2. Future Works 81 In Chapter 3, we developed a generic analytical framework for OFDM/F-OFDM systems to address the 5G NR numerology coexistence issue in which the process of multiplexing different numerologies has been formulated. Based on the framework developed in Chapter 3, the analytical expressions of the inband interferences, including ICI, forward ISI, and backward ISI, were derived, and the interference-free condition was developed accordingly. In addition, a low-complexity FEQ algorithm - BwPIC was proposed to mitigate the IntraNI. Furthermore, the effect of FFRS to single antenna and multi-antenna f-OFDM systems was investigated, and a pre-equalization approach was proposed to tackle it. The inter-numerology interference was analyzed based on the model developed in Chapter 3. The analytical metric for quantifying the level of the distortion was derived as a function of several system parameters. The impact of these parameters on the internumerology interference were examined both analytically and numerically. The usages of analytical expression inter-numerology interference are discussed. A case study on optimizing power allocation based on the offered analysis was also presented. The work conducted in this thesis provides an analytical guidance on the system design in support of 5G multi-service transmission over a unified physical infrastructure. 6.2 Future Works In the thesis, a frame work for multiplexing different numerology over a carrier has been developed which provides many future research opportunities including (but not limited): 1. Joint guard band and guard interval optimization: various slot configurations and user equipment (UE) scheduling guidelines reveal that few restrictions exist regarding scheduling users in time domain. This implies that the guard times can also be utilized flexibly, similar to guard bands. Combining time-frequency guard flexibility yields flexible placement of the empty resource elements. 82 6.2. Future Works 2. Optimal filter design to minimize the net interference in multi-numerology system: additional filter processing can be introduced in multi-numerology systems to mitigate the interference between different numerologies. However, this also leads to some level of interference within each numerology. The employed filter has a significant influence on the net interference. For example, a longer filter has a better localization in the frequency domain but worse localization in the time domain, which suggests a lower Inter-NI and higher Intra-NI. Moreover, other parameter such as roll-off factor also affects the time/frequency localization. Therefore an optimized filter balancing the Inter-NI and Intra-NI is desired to achieve a minimum net interference. 3. Joint optimization on bandwidth part, subcarrier and power allocation in the presence of multiple numerologies: the global solution on spectrum/energy efficiency can be achieved by jointly optimizing of bandwidth part, subcarrier and power. We have showed a use case on optimizing power in this thesis, which serves as a starting point to capitalize on the capabilities that the analytical framework enables. This work can be naturally extended to more complicated scenarios. 4. User-specific numerology adaption based on communication environments and services: smaller numerologies are beneficial for high Doppler and low latency scenarios, while bigger numerologies are attractive for extensive coverage and longer dispersive channels. When the communication environment or the service of a user changes, the numerology should be adapted to achieve a better quality of experience, at the same time bring other benefits such as reduced power consumption or better spectrum efficiency. 5. Optimization on number of active numerologies: the efficient number of active numerologies can be simultaneously employed by users. The algorithm aims to minimize various overheads to provide a practical solution satisfying different service and user requirements using multi-numerology structures. All different numerologies that are defined in standards do not need to be used in every situation. Basically, the amount of total guard band in the lattice increases with increasing number of numerologies. Hence, there is a trade-off between the spec- 6.2. Future Works 83 tral efficiency and multi-numerology system flexibility. 6. System performance for multi-numerology systems in the presence of doublydispersive channel : the system performance discussed in this thesis are based on the assumption of perfect synchronization both in the time domain and in the frequency domain. The effect from doubly-dispersive channel to the multinumerology systems is one of interesting direction worth to investigate in the future. Appendix A Proof of Proposition 3.1 and 3.2 As shown in Fig. 3.3, the length of symbols is different across numerologies. The symbol length (including CR) of the i-th numerology can be calculated as Lref , ν (i) L(i) = (A.1) where Lref is the symbol length of 15 kHz subcarrier spacing. The k-th symbol of the ref ref i-th numerology spans in the time interval k Lν (i) , (k + 1) Lν (i) . When the i-th numerology is multiplexed with the j-th numerology, the later has greater or shorter symbol length depends on its subcarrier spacing. (i−) If j ∈ Snum , i.e., the j-th numerology has a narrower subcarrier spacing and a greater ref ref , (k + 1) Lν (i) occupies symbol length. Therefore, the signal in the time interval k Lν (i) only a portion of a OFDM window of the j-th numerology, and the index of the window can be calculated as $ kLref /ν (i) Lref /ν (j) % $ % ν (j) = k (i) . ν Moreover, the symbol of the j-th numerology is ν (i) ν (i) (A.2) times long as that of the i-th numerology. If we equally divide each OFDM symbol of the j-th numerology into ν (i) ν (j) ref ref symbol parts (SPs), then exactly one of them fits in the time interval k Lν (i) , (k+1) Lν (i) , and it can be found as ! j k k ν (j) j ν (j) k ν (i) (i) (j) = k − ν /ν =k k (i) − k (i) ν ν ν (j) ν (i) /ν (j) 84 mod ν (i) /ν (j) . (A.3) 85 To sum up, the (k mod ν (i) )-th ν (j) (j) SP of the (bk νν (i) c)-th OFDM symbol of the j-th (i−) numerology overlaps with k-th OFDM symbol of the i-th numerology if j ∈ Snum . (i+) In contrast, if j ∈ Snum , each symbol length of the i-th numerology equals the sum of corresponding ν (j) ν (i) symbols of the j-th numerology. The Index of the first of those ref ref symbol in the time interval k Lν (i) , (k + 1) Lν (i) can be calculated as ν (j) kLref /ν (i) = k , Lref /ν (j) ν (i) (A.4) and the index of the last one can be obtained as (k + 1)Lref /ν (i) ν (j) − 1 = (k + 1) − 1. Lref /ν (j) ν (i) Therefore, the k-th symbol of the i-th numerology overlaps with j-th numerology in the range of kν (j) ν (i) , kν (j) ν (i) + 1, · · · , (k+1)ν (j) ν (i) (A.5) ν (j) ν (i) symbols of the (i+) − 1 if j ∈ Snum . Appendix B (i) Derivation of zk (i) Substituting the expression of rk in (3.25) into (3.31), we obtain (i) (i) (i) zk =V(i:u) H(i:u) sk−2 + V(i:u) H(i:m) + V(i:m) H(i:u) sk−1 (i) (i) + V(i:m) H(i:m) + V(i:l) H(i:u) sk + V(i:l) H(i:m) sk+1 . (B.1) Since the multiplication of two strict upper triangular matrices results in a zero matrix, (i) the term V(i:u) H(i:u) s<i> k−2 can be canceled. Replacing sk of the above equation with (i) its expression in (3.7), followed by merging of similar items with respect to xk , yields (i) (i) zk =V(i:u) H(i:m) U(i:u) xk−2 (i) + V(i:u) H(i:m) U(i:m) + V(i:m) H(i:u) U(i:m) + V(i:m) H(i:m) U(i:u) + V(i:l) H(i:u) U(i:u) xk−1 (i) + V(i:u) H(i:m) U(i:l) + V(i:m) H(i:u) U(i:l) + V(i:m) H(i:m) U(i:m) + V(i:l) H(i:m) U(i:u) xk (i) + V(i:m) H(i:m) U(i:l) + V(i:l) H(i:u) U(i:l) + V(i:l) H(i:m) U(i:m) xk+1 (i) +V(i:l) H(i:m) U(i:l) xk+2 . (B.2) It is proved in appendix D that U(i:u) H(i:m) is a strict upper triangular matrix, and V(i:l) H(i:m) is a strict lower triangular matrix. As the multiplication of two strict upper/lower triangular matrices equals a zero matrix, we have V(i:u) H(i:m) U(i:u) = 0, (i) V(i:l) H(i:m) U(i:l) = 0 , and V(i:l) H(i:u) U(i:u) = 0. zk in (B.2) can then be simplified as (i) (i) (i) (i) (i) (i) zk = Θ(i) pre xk−1 + Θ xk + Θnext xk+1 , 86 87 (i) where Θpre = V(i:u) H(i:m) U(i:m) + V(i:m) H(i:u) U(i:m) + V(i:m) H(i:m) U(i:u) , Θ(i) = V(i:u) H(i:m) U(i:l) + V(i:m) H(i:u) U(i:l) + V(i:m) H(i:m) U(i:m) + V(i:l) H(i:m) U(i:u) , (i) Θnext = V(i:m) H(i:m) U(i:l) + V(i:l) H(i:u) U(i:l) + V(i:l) H(i:m) U(i:m) . Appendix C (i) Derivation of z̃k (i) Substituting the expression of rk in (3.25) into (3.32) yields (i) (i) z̃k = V(i:u) H(i:m) + V(i:m) H(i:u) s̃k−1 (i) (i) + V(i:m) H(i:m) + V(i:l) H(i:u) s̃k + V(i:l) H(i:m) s̃k+1 . (C.1) (i) According to Eq. (3.17), the vector s̃k is the mixed signal from all numerologies except (i) the i-th one. Thus, we can express z̃k as a sum of the signal from those numerologies as (i) z̃k = X (i←j) zk , (C.2) j∈Snum \{i} (i←j) where zk can be interpreted as the signal from the j-th numerology which lies in the k-th OFDM window of the i-th numerology. Based on (C.1), it can be expressed as (i←j) zk (i←j) = V(i:u) H(i:m) + V(i:m) H(i:u) sk−1 (i←j) (i←j) + V(i:m) H(i:m) + V(i:l) H(i:u) sk + V(i:l) H(i:m) sk+1 , (i←j) Where the definition of sk (i+) (C.3) (i−) is given in proposition 3.1 and proposition 3.2 for j ∈ Snum (i←j) and j ∈ Snum , respectively. In the following, we will further expand zk (i+) and j ∈ Snum , respectively. 88 (i−) for j ∈ Snum 89 (i−) In the case of j ∈ Snum C.0.1 (i←j) After some basic algebraic manipulations based on (3.7) and the expression of sk (i←j) given in proposition 3.1, we can express zk in (C.3) as a sum of a dominant term and many trivial terms as (i←j) zk (i←j) (i←j) = V(i:m) H(i:m) Ck U(j:m) xk | {z + } dorminate term (i←j) k , | {z } (C.4) trivial terms with (i←j) k (i←j) (i←j) (i←j) = V(i:u) H(i:m) Ck−1 U(j:u) + V(i:m) H(i:u) Ck−1 U(j:u) xk−1 (i←j) (i←j) (i←j) + V(i:u) H(i:m) Ck−1 U(j:m) + V(i:m) H(i:u) Ck−1 U(j:m) xk (i←j) (i←j) (i←j) + V(i:u) H(i:m) Ck−1 U(j:l) + V(i:m) H(i:u) Ck−1 U(j:l) xk+1 (i←j) (j:u) (i←j) (j:u) (i←j) + V(i:m) H(i:m) Ck U + V(i:l) H(i:u) Ck U xk−1 (i←j) (j:l) (i←j) (j:l) (i←j) + V(i:m) H(i:m) Ck U + V(i:l) H(i:u) Ck U xk+1 (i←j) + V(i:l) H(i:u) Ck (i←j) U(j:m) xk (i←j) (i←j) + V(i:l) H(i:m) Ck+1 U(j:m) xk (i←j) where xk (j) = x (j) k ν (i) (j) , x(i←j) = x k (i←j) (i←j) + V(i:l) H(i:m) Ck+1 U(j:l) xk+1 , (j) (k−1) ν (i) ν (i←j) (i←j) + V(i:l) H(i:m) Ck+1 U(j:u) xk−1 (j) , and x(i←j) = x k (j) (k+1) ν (i) ν (C.5) . ν As the trivial terms correspond to filter and channel spreadings, and their energy is significantly less than that of the main body of signal. Moreover, assume that CR is sufficiently longer to capture the main lob of the filters and the channel spreading. (i←j) Thus, the residual spreading can be ignored, and we can approximate zk (i←j) zk C.0.2 (i←j) ≈ V(i:m) H(i:m) Ck (i←j) U(j:m) xk , j ∈ S(i−) num . as (C.6) (i+) In the case of j ∈ Snum (i←j) Based on (3.7) and the expression of sk procedure in A ( j ∈ (i←j) zk (i−) Snum ), given in proposition 3.2, following the similar (i←j) we can approximate zk (i+) (j ∈ Snum ) as ν (j) (i←j) ≈ V(i:m) H(i:m) blkdiag U(j:m) , (i) xk , j ∈ S(i+) num . ν (C.7) 90 " where (i←j) xk = T (j) x kν (j) , x kν (j) ν (i) (i←j) Combining zk (i) z̃k ≈ T (j) ν (i) +1 (i−) T (j) , · · · , x (k+1)ν (j) ν (i) #T . −1 (i+) for j ∈ Snum and j ∈ Snum , we finally obtain X j∈Snum \{i} (i←j) zk = (i←j) X V(i:m) H(i:m) Ck (i←j) U(j:m) xk (i−) j∈Snum + ν (j) (i←j) V(i:m) H(i:m) blkdiag U(i:m) , (i) xk . ν (i+) X j∈Snum (C.8) Appendix D The Proof of Strictly Triangular Matrices The product of U(i:u) /V(i:u) and H(i:m) is a strictly upper triangular matrix, such as (i) (i) (i) Dr,c = 0, 0 ≤ ∀r, ∀c ≤ L(i) , when r > c − [L(i) − N2u − (Nch − 1)]. As U(i:u) and V(i:u) are matrices with the same structure, we only give detailed steps to prove one of them, the other one can be conducted in the similar fashion. (i) (i) Dr,c = L X (i:u) (i:m) Ur,k Hk,c k=1 (i) c+Nch −1 = X (i) (i:u) (i:m) Ur,k Hk,c k=1 Based on the condition r > c − L(i) − + L X (i:u) (i:m) Ur,k Hk,c . (D.1) (i) k=c+Nch (i) Nu 2 (i) (i) − (Nch − 1) , when 1 ≤ k ≤ c + Nch − 1, we have (i) k− L (i) (i) (i) Nu Nu Nu (i) (i) (i) − ≤ k ≤ c + Nch − 1 − L − <r ⇒k <r+L − , 2 2 2 (D.2) (i:u) and this gives Ur,k = 0 according to (3.28). Therefore, the first term of (D.1), (i) Pc+Nch −1 (i:u) (i:m) Ur,k Hk,c = 0. k=1 (i) (i:m) When c + Nch ≤ k ≤ L(i) , we have Hk,c 91 = 0 based on (3.21), then the second 92 (i) term of (D.1), L X (i:u) (i:m) Ur,k Hk,c (i) (i) = 0, is also proved. Therefore, Dr,c = 0, 0 ≤ k=j+Nch ∀r, ∀c ≤ L(i) is proved because both its sum terms in (D.1) equals to zero, when (i) (i) r > c − L(i) − N2u − (Nch − 1) . 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